$V$ and $W$ are finite dimensional vector spaces over $k$. I need a basis free isomorphism between $V^*\otimes_{k} W^*$ and $Bil_{k}(V\times W,k)$.
My attempt: We have a bilinear map $V^*\times W^*\rightarrow Bil_{k}(V\times W,k)$ sending $(f,g)\mapsto \Big((v,w)\mapsto f(v)g(w)\Big)$. This induces a linear map from $V^*\otimes W^*\rightarrow Bil_{k}(V\times W,k)$, sending $f\otimes g\mapsto \Big((v,w)\mapsto f(v)g(w)\Big)$. I need a $k$-linear map from the other direction so that the compositions will be identity.