Showing that the set of vectors $e_k-e_l$ spans $ V=\left\{ (a_1,...,a_n) \ | \ a_1+\dots + a_n=0 \right\}. $

Question

As the title says, I want to show that the set of vectors $$\{ e_k-e_l \ | \ k, l \in \{1, \dots, n \} \}$$ spans the vector space given by $$ V=\left\{ (a_1,...,a_n) \ | \ a_1+\dots + a_n=0 \right\}. $$ That means that I should show that any $v=\sum_i a_ie_i\in V$ can be written as a linear combination of the vectors $e_k-e_l$. Consider: $$ \sum_i a_i e_i=\sum_i a_i e_i-\left(\sum_{i} a_i\right) e_j=\sum_i a_i\left(e_i-e_j\right). $$ Is this sufficient (and correct)?