Let $V$ be a vector space on a field $\mathbb{K}$. Let $(V^\ast)^{\otimes q}$ be the space of $q$-linear forms, $\Lambda^q(V^\ast)$ the space of skew-symmetric multilinear forms and $Sym^q(V^\ast)$ the space of symmetric multilinear forms.
I proved that $$ (V^\ast)^{\otimes 2}=\Lambda^2(V^\ast)\oplus Sym^2(V^\ast) $$ by using the isomorphism $(V^\ast)^{\otimes 2}\cong gl(\mathbb{K},n)$.
I'd like to generalize for $q\ge 3$, but I don't have any ideas