Let $T$ be an endomorphism of a finite-dimensional vector space $V$. Let $$f(x)=x^n+c_1x^{n-1}+ \dots + c_n$$ be the characteristic polynomial of $T$. It is well known that $c_m=(-1)^m\text{tr}(\bigwedge^mT)$.
If the base field is $\mathbf{C}$, then we can prove it using a density argument. The statement is true for diagonalizable matrices, which are dense in $M_n(\mathbf{C})$. This actually enough to prove it general, but I don't find it very illuminating. I would like to see an abstract proof of this result.
Thank you!