I've been thinking about this for a while and I've found that for the 2x2 case the matrices are just $\{e_{11}, e_{12} + e_{21}, e_{21}, e_{22}\}$ where $e_{ij}$ is the matrix with $1$ on the $ij$ entry, $0$'s everywhere else. However I can't seem to see any patterns with higher orders. I need to find the basis for the $n \times n$ case.
Any help is appreciated.