Defining the trace in the usual way as a function $Tr: F^{n\times n} \rightarrow F$, where $F$ is some field. I want to show that $\text{ker}(Tr)=\text{span}_F(\{AB-BA|A,B\in F^{n\times n}\})$.
So far I've certainly deduced that $\text{im}(Tr)=F$, and so $\text{dim}(\text{ker}(Tr))=n^2-1$. Now I need only find a basis for $S$ (which I'll use to denote the spanning set above), since clearly $S\subseteq \text{ker}(Tr)$.
I've fooled around writing $A$ and $B$ as linear combinations of the standard $E_{ij}$, and messing around with the indices to find redundancies etc, but I can't seem to make any headway. Am I missing some clever trick. Perhaps I need to rewrite the $E_{ij}$ in some more useful way?