$\mathfrak{sl}(n,\mathbb{C})$ acts transitively on $\mathbb{C}^n$.

Question

I want to show that the Lie algebra of all trace-$0$ complex matrices acts transitively on $\mathbb{C}^n$, so the standard representation is irreducible. Let $v=\sum_i a_ie_i$, then the matrix $$ A_{i1}=a_i, A_{22}=-a_1 $$ has trace $0$ and maps $e_1$ to $v$. Is this a correct argument for the irreducibility of the standard representation of $\mathfrak{sl}(n,\mathbb{C})$ (defined via $X.v=Xv$)?