Coefficient of $x^{n-2}$ in $(x-1)(x-2)(x-3)\dotsm(x-n)$ Question

Find the coefficient of $x^{n-2}$  in the expression $$(x-1)(x-2)(x-3)\dots(x-n)~~.$$

My approach

The coefficient of $x^n$ is $1$. The coefficient of  $x^{n-1}$ is $- \frac{n(n+1)}{2}$

But I cannot proceed from here.
I would appreciate any help.

Finding the coefficient of $x^{n-2}$ requires picking $2$ terms from the product to multiply the constants. Thus, we get the coefficient to be
$$
\begin{align}
\sum_{k=2}^n\sum_{j=1}^{k-1}jk
&=\sum_{k=2}^n\sum_{j=1}^{k-1}\binom{j}{1}k\\
&=\sum_{k=2}^n\binom{k}{2}k\\
&=\sum_{k=2}^n\binom{k}{2}((k-2)+2)\\
&=\sum_{k=2}^n\left(3\binom{k}{3}+2\binom{k}{2}\right)\\
&=3\binom{n+1}{4}+2\binom{n+1}{3}\\[3pt]
&=\frac{(3n+2)(n^3-n)}{24}
\end{align}
$$
