Matrix-unit basis of the unitary Lie algebra ($i{E}_{ii},E_{ij}-E_{ji},i(E_{ij}+E_{ji})$)

A real basis of the unitary Lie algebra is $i{E}_{ii}$, $E_{ij}-E_{ji}$ and $i(E_{ij}+E_{ji})$ for $i<j$, where $E_{ij}$ are matrix units. These separately span the imaginary diagonal and real and imaginary upper off-diagonal entries.