For \(\displaystyle i \neq j\), the elementary matrices \(\displaystyle E_{i j}\) , \(\displaystyle (1\) in the \(\displaystyle i,j \) position and \(\displaystyle 0\) everywhere else)

can be written in the required form since

\(\displaystyle E_{i j}=E_{i 1}E_{1j}-E_{1 j}E_{i 1}\)

Any diagonal matrix whose trace is \(\displaystyle 0\) can be written as a linear combination of matrices of the form

\(\displaystyle E_{1 1}-E_{i i}\) where \(\displaystyle i \geq 2\)