I want to find out what $e^{E_{ij}}$ equals, where $E_{ij}$ is the $(i,j)$-th matrix in the standard basis of $M_n(\mathbb C)$.
It seems that in the case that $i\ne j$ we have that: $$e^{E_{ij}}= I + \frac{E_{ij}}{1}+0+\cdots = I + E_{ij}$$
and in the case that $i=j$ we find that: \begin{align} E_{ii}^k=E_{ii}&\implies e^{E_{ii}}=I + E_{ii}+\frac{E_{ii}}{2}+\cdots\\ &\implies e^{E_{ii}}= I+\sum_{n=1}^\infty \frac{E_{ii}}{n!}\\ &\implies e^{E_{ii}}=(e-1)E_{ii}+I. \end{align}
Is that correct?