The question is on the title ($n \in \mathbb{N}^*$). To be clear, unitary basis here means basis consisting of (complex) unitary matrices.
I wonder this question because recently I've read about Pauli matrices, which are all unitary and, together with the identity matrix $I_2$, make up a basis of the $\mathbb{R}$-vector space of $2 \times 2$ Hermitian matrices. There are 4 elements in that basis, suitably as the vector space has $\dim = 2 \times 2 = 4$. In the general case, this would be $n^2$.
Disclaimer: I should prove the existence of such basis first, but I haven't done it. Still I think there exists at least one $\forall n$.