To understand what you are doing you need to dance along with all those many concepts and equations. There is no shortcut there.
If you are asking about an explicit expression for the local stiffness matrix for a trilinear element for elasticity, I would say that it might be unpractical.
The stiffness matrix is computed by the following integral
$$ K = \int\limits_{[-1,1]^3} B^T C B\, |J| \mathrm{d}r\, \mathrm{d}s\, \mathrm{d}t$$
with $C$ the stiffness tensor in Voigt notation, and $B$ the displacement-to-strain matrix. The 3 columns related to node $i$ of $B$ is of the form
$$B = \begin{bmatrix}
\frac{\mathrm{d} N_i}{\mathrm{d}x} & 0 & 0\\
0 &\frac{\mathrm{d} N_i}{\mathrm{d}y} & 0\\
0 & 0 &\frac{\mathrm{d} N_i}{\mathrm{d}z}\\
0 &\frac{\mathrm{d} N_i}{\mathrm{d}z} & \frac{\mathrm{d} N_i}{\mathrm{d}y}\\
\frac{\mathrm{d} N_i}{\mathrm{d}z} &0 & \frac{\mathrm{d} N_i}{\mathrm{d}x}\\
\frac{\mathrm{d} N_i}{dy} &\frac{\mathrm{d} N_i}{\mathrm{d}x} & 0
\end{bmatrix}$$
the remaining columns are similar, just change the index $i$. The indices correspond to the interpolation functions:
$$N = \begin{bmatrix}
\frac{1}{8} \left(- r + 1\right) \left(- s + 1\right) \left(t + 1\right)\\
\frac{1}{8} \left(r + 1\right) \left(- s + 1\right) \left(t + 1\right)\\
\frac{1}{8} \left(r + 1\right) \left(s + 1\right) \left(t + 1\right)\\
\frac{1}{8} \left(- r + 1\right) \left(s + 1\right) \left(t + 1\right)\\
\frac{1}{8} \left(- r + 1\right) \left(- s + 1\right) \left(- t + 1\right)\\
\frac{1}{8} \left(r + 1\right) \left(- s + 1\right) \left(- t + 1\right)\\
\frac{1}{8} \left(r + 1\right) \left(s + 1\right) \left(- t + 1\right)\\
\frac{1}{8} \left(- r + 1\right) \left(s + 1\right) \left(- t + 1\right)
\end{bmatrix}$$
Notice that the derivatives in $B$ are in $(x, y, z)$ and the interpolation functions are in $(r, s, t)$. And you would need to compute the chain derivates.
If we express the stiffness tensor as
$$C = \lambda C_1 + \mu C_2$$
with
$$C_1 = \begin{bmatrix}
1 & 1 & 1 & 0 & 0 & 0\\
1 & 1 & 1 & 0 & 0 & 0\\
1 & 1 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}$$
and
$$C_2 = \begin{bmatrix}
2 & 0 & 0 & 0 & 0 & 0\\
0 & 2 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}$$
And you split your stiffness matrix as $K = K_1 + K_2$, with
$$\frac{288 K_1}{L}= \begin{bmatrix}
8 & 6 & -6 & -8 & 6 & -6 & -4 & -6 & -3 & 4 & -6 & -3 & 4 & 3 & 6 & -4 & 3 & 6 & -2 & -3 & 3 & 2 & -3 & 3\\
6 & 8 & -6 & -6 & 4 & -3 & -6 & -4 & -3 & 6 & -8 & -6 & 3 & 4 & 6 & -3 & 2 & 3 & -3 & -2 & 3 & 3 & -4 & 6\\
-6 & -6 & 8 & 6 & -3 & 4 & 3 & 3 & 2 & -3 & 6 & 4 & -6 & -6 & -8 & 6 & -3 & -4 & 3 & 3 & -2 & -3 & 6 & -4\\
-8 & -6 & 6 & 8 & -6 & 6 & 4 & 6 & 3 & -4 & 6 & 3 & -4 & -3 & -6 & 4 & -3 & -6 & 2 & 3 & -3 & -2 & 3 & -3\\
6 & 4 & -3 & -6 & 8 & -6 & -6 & -8 & -6 & 6 & -4 & -3 & 3 & 2 & 3 & -3 & 4 & 6 & -3 & -4 & 6 & 3 & -2 & 3\\
-6 & -3 & 4 & 6 & -6 & 8 & 3 & 6 & 4 & -3 & 3 & 2 & -6 & -3 & -4 & 6 & -6 & -8 & 3 & 6 & -4 & -3 & 3 & -2\\
-4 & -6 & 3 & 4 & -6 & 3 & 8 & 6 & 6 & -8 & 6 & 6 & -2 & -3 & -3 & 2 & -3 & -3 & 4 & 3 & -6 & -4 & 3 & -6\\
-6 & -4 & 3 & 6 & -8 & 6 & 6 & 8 & 6 & -6 & 4 & 3 & -3 & -2 & -3 & 3 & -4 & -6 & 3 & 4 & -6 & -3 & 2 & -3\\
-3 & -3 & 2 & 3 & -6 & 4 & 6 & 6 & 8 & -6 & 3 & 4 & -3 & -3 & -2 & 3 & -6 & -4 & 6 & 6 & -8 & -6 & 3 & -4\\
4 & 6 & -3 & -4 & 6 & -3 & -8 & -6 & -6 & 8 & -6 & -6 & 2 & 3 & 3 & -2 & 3 & 3 & -4 & -3 & 6 & 4 & -3 & 6\\
-6 & -8 & 6 & 6 & -4 & 3 & 6 & 4 & 3 & -6 & 8 & 6 & -3 & -4 & -6 & 3 & -2 & -3 & 3 & 2 & -3 & -3 & 4 & -6\\
-3 & -6 & 4 & 3 & -3 & 2 & 6 & 3 & 4 & -6 & 6 & 8 & -3 & -6 & -4 & 3 & -3 & -2 & 6 & 3 & -4 & -6 & 6 & -8\\
4 & 3 & -6 & -4 & 3 & -6 & -2 & -3 & -3 & 2 & -3 & -3 & 8 & 6 & 6 & -8 & 6 & 6 & -4 & -6 & 3 & 4 & -6 & 3\\
3 & 4 & -6 & -3 & 2 & -3 & -3 & -2 & -3 & 3 & -4 & -6 & 6 & 8 & 6 & -6 & 4 & 3 & -6 & -4 & 3 & 6 & -8 & 6\\
6 & 6 & -8 & -6 & 3 & -4 & -3 & -3 & -2 & 3 & -6 & -4 & 6 & 6 & 8 & -6 & 3 & 4 & -3 & -3 & 2 & 3 & -6 & 4\\
-4 & -3 & 6 & 4 & -3 & 6 & 2 & 3 & 3 & -2 & 3 & 3 & -8 & -6 & -6 & 8 & -6 & -6 & 4 & 6 & -3 & -4 & 6 & -3\\
3 & 2 & -3 & -3 & 4 & -6 & -3 & -4 & -6 & 3 & -2 & -3 & 6 & 4 & 3 & -6 & 8 & 6 & -6 & -8 & 6 & 6 & -4 & 3\\
6 & 3 & -4 & -6 & 6 & -8 & -3 & -6 & -4 & 3 & -3 & -2 & 6 & 3 & 4 & -6 & 6 & 8 & -3 & -6 & 4 & 3 & -3 & 2\\
-2 & -3 & 3 & 2 & -3 & 3 & 4 & 3 & 6 & -4 & 3 & 6 & -4 & -6 & -3 & 4 & -6 & -3 & 8 & 6 & -6 & -8 & 6 & -6\\
-3 & -2 & 3 & 3 & -4 & 6 & 3 & 4 & 6 & -3 & 2 & 3 & -6 & -4 & -3 & 6 & -8 & -6 & 6 & 8 & -6 & -6 & 4 & -3\\
3 & 3 & -2 & -3 & 6 & -4 & -6 & -6 & -8 & 6 & -3 & -4 & 3 & 3 & 2 & -3 & 6 & 4 & -6 & -6 & 8 & 6 & -3 & 4\\
2 & 3 & -3 & -2 & 3 & -3 & -4 & -3 & -6 & 4 & -3 & -6 & 4 & 6 & 3 & -4 & 6 & 3 & -8 & -6 & 6 & 8 & -6 & 6\\
-3 & -4 & 6 & 3 & -2 & 3 & 3 & 2 & 3 & -3 & 4 & 6 & -6 & -8 & -6 & 6 & -4 & -3 & 6 & 4 & -3 & -6 & 8 & -6\\
3 & 6 & -4 & -3 & 3 & -2 & -6 & -3 & -4 & 6 & -6 & -8 & 3 & 6 & 4 & -3 & 3 & 2 & -6 & -3 & 4 & 6 & -6 & 8
\end{bmatrix}$$
and
$$ \frac{288 K_2}{L} = \begin{bmatrix}
32 & 6 & -6 & -8 & -6 & 6 & -10 & -6 & 3 & 4 & 6 & -3 & 4 & 3 & -6 & -10 & -3 & 6 & -8 & -3 & 3 & -4 & 3 & -3\\
6 & 32 & -6 & 6 & 4 & -3 & -6 & -10 & 3 & -6 & -8 & 6 & 3 & 4 & -6 & 3 & -4 & -3 & -3 & -8 & 3 & -3 & -10 & 6\\
-6 & -6 & 32 & -6 & -3 & 4 & -3 & -3 & -4 & -3 & -6 & 4 & 6 & 6 & -8 & 6 & 3 & -10 & 3 & 3 & -8 & 3 & 6 & -10\\
-8 & 6 & -6 & 32 & -6 & 6 & 4 & -6 & 3 & -10 & 6 & -3 & -10 & 3 & -6 & 4 & -3 & 6 & -4 & -3 & 3 & -8 & 3 & -3\\
-6 & 4 & -3 & -6 & 32 & -6 & 6 & -8 & 6 & 6 & -10 & 3 & -3 & -4 & -3 & -3 & 4 & -6 & 3 & -10 & 6 & 3 & -8 & 3\\
6 & -3 & 4 & 6 & -6 & 32 & 3 & -6 & 4 & 3 & -3 & -4 & -6 & 3 & -10 & -6 & 6 & -8 & -3 & 6 & -10 & -3 & 3 & -8\\
-10 & -6 & -3 & 4 & 6 & 3 & 32 & 6 & 6 & -8 & -6 & -6 & -8 & -3 & -3 & -4 & 3 & 3 & 4 & 3 & 6 & -10 & -3 & -6\\
-6 & -10 & -3 & -6 & -8 & -6 & 6 & 32 & 6 & 6 & 4 & 3 & -3 & -8 & -3 & -3 & -10 & -6 & 3 & 4 & 6 & 3 & -4 & 3\\
3 & 3 & -4 & 3 & 6 & 4 & 6 & 6 & 32 & 6 & 3 & 4 & -3 & -3 & -8 & -3 & -6 & -10 & -6 & -6 & -8 & -6 & -3 & -10\\
4 & -6 & -3 & -10 & 6 & 3 & -8 & 6 & 6 & 32 & -6 & -6 & -4 & -3 & -3 & -8 & 3 & 3 & -10 & 3 & 6 & 4 & -3 & -6\\
6 & -8 & -6 & 6 & -10 & -3 & -6 & 4 & 3 & -6 & 32 & 6 & 3 & -10 & -6 & 3 & -8 & -3 & -3 & -4 & 3 & -3 & 4 & 6\\
-3 & 6 & 4 & -3 & 3 & -4 & -6 & 3 & 4 & -6 & 6 & 32 & 3 & -6 & -10 & 3 & -3 & -8 & 6 & -3 & -10 & 6 & -6 & -8\\
4 & 3 & 6 & -10 & -3 & -6 & -8 & -3 & -3 & -4 & 3 & 3 & 32 & 6 & 6 & -8 & -6 & -6 & -10 & -6 & -3 & 4 & 6 & 3\\
3 & 4 & 6 & 3 & -4 & 3 & -3 & -8 & -3 & -3 & -10 & -6 & 6 & 32 & 6 & 6 & 4 & 3 & -6 & -10 & -3 & -6 & -8 & -6\\
-6 & -6 & -8 & -6 & -3 & -10 & -3 & -3 & -8 & -3 & -6 & -10 & 6 & 6 & 32 & 6 & 3 & 4 & 3 & 3 & -4 & 3 & 6 & 4\\
-10 & 3 & 6 & 4 & -3 & -6 & -4 & -3 & -3 & -8 & 3 & 3 & -8 & 6 & 6 & 32 & -6 & -6 & 4 & -6 & -3 & -10 & 6 & 3\\
-3 & -4 & 3 & -3 & 4 & 6 & 3 & -10 & -6 & 3 & -8 & -3 & -6 & 4 & 3 & -6 & 32 & 6 & 6 & -8 & -6 & 6 & -10 & -3\\
6 & -3 & -10 & 6 & -6 & -8 & 3 & -6 & -10 & 3 & -3 & -8 & -6 & 3 & 4 & -6 & 6 & 32 & -3 & 6 & 4 & -3 & 3 & -4\\
-8 & -3 & 3 & -4 & 3 & -3 & 4 & 3 & -6 & -10 & -3 & 6 & -10 & -6 & 3 & 4 & 6 & -3 & 32 & 6 & -6 & -8 & -6 & 6\\
-3 & -8 & 3 & -3 & -10 & 6 & 3 & 4 & -6 & 3 & -4 & -3 & -6 & -10 & 3 & -6 & -8 & 6 & 6 & 32 & -6 & 6 & 4 & -3\\
3 & 3 & -8 & 3 & 6 & -10 & 6 & 6 & -8 & 6 & 3 & -10 & -3 & -3 & -4 & -3 & -6 & 4 & -6 & -6 & 32 & -6 & -3 & 4\\
-4 & -3 & 3 & -8 & 3 & -3 & -10 & 3 & -6 & 4 & -3 & 6 & 4 & -6 & 3 & -10 & 6 & -3 & -8 & 6 & -6 & 32 & -6 & 6\\
3 & -10 & 6 & 3 & -8 & 3 & -3 & -4 & -3 & -3 & 4 & -6 & 6 & -8 & 6 & 6 & -10 & 3 & -6 & 4 & -3 & -6 & 32 & -6\\
-3 & 6 & -10 & -3 & 3 & -8 & -6 & 3 & -10 & -6 & 6 & -8 & 3 & -6 & 4 & 3 & -3 & -4 & 6 & -3 & 4 & 6 & -6 & 32
\end{bmatrix}$$
In both cases, the matrix is lacking the factor to change coordinates $L/8$ and are multiplied by 36 to make numbers prettier. Still, after all this cosmetic manipulation of the computation, it is not practical to use these $24\times 24$ matrices explicitly.