Let $V$ be a $k$-vector space. Is there a way to express $T_n(V)$ (n-fold tensor of $V$) in terms of symmetric products $Sym_i(V)$ and exterior products $\Lambda^j(V)$ ?
Maybe this is not a very well-defined question, but I just want to know the results and references. As commented below, there seems to be a way of expressing $T_n(V)$ as direct sums.