Let $\cal M$$_n$ be the vector space of all $n\times n$-matrices over an algebraically closed field $\cal F$ of characteristic zero. We describe the linear span of the conjugacy class (with respect to the full linear group $\cal{GL}_n$) of an arbitrary matrix $A\in\cal M_n$, and derive the existence of some particular bases of $\cal M_n$. Moreover, we propose certain observations on finite sequences of nilpotent matrices and their linear spans.