Is $\mathfrak{s}\mathfrak{u}(d) \otimes \mathfrak{s}\mathfrak{u}(d)=\mathfrak{s}\mathfrak{u}(d^2)$?

Question

Given a basis $a_i$ for the lie algebra $\mathfrak{s}\mathfrak{u}(d)$, does the set of elements $a_i \otimes a_j$ form a basis for $\mathfrak{s}\mathfrak{u}(d^2)$?