'elementary' proof that all matrix with zero trace has the form $AB-BA$

Question

How to prove the fact mentioned in the title without using the theory of lie algebras?

For the record, though questions about this fact has indeed appeared in MSE several times, yet up til now, I think no one has given any 'elementary'(only use linear algebra) proof of this fact on MSE. So it's not a duplicate.

Any reference or answer would be greatly appreciated. Thanks in advance.

P.S. this question is not equivalent to this:

If $V_0$ is the subspace of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace} (A)=0\}$

or this: Dimensionality of null space when Trace is Zero