Let $f\in K[V]_d$ be a homogeneous polynomial of degree $d$ on the vector space $V$ over the field $K$ where char$(K)=0$. It is well-known that there is an isomorphism $$ \pi : K[V]_d\rightarrow Sym^d(V^*)\subseteq (V^*)^{\otimes m} $$ i.e. $f$ can be seen as a multilinear function $$ f:\times_{i=1}^d V\rightarrow K$$
Now assume that $V=W^{\oplus m}$. From the coordinate description of the polarization, it seems like $f$ can be further seen as a multilinear function $$ f: \times_{i=1}^{md} W\rightarrow K$$ However I cannot write this map without coordinates.
Question : Is there a natural embedding $K[W^{\oplus m}]_d\rightarrow (W^*)^{\otimes dm}$