Let $V_1, \cdots, V_n$ be vector spaces over $k$; let $L(V_1, \cdots, V_n)$ be the vector space of multilinear maps from $V_1 \times \cdots \times V_n$ to $k$.
we can identity elements of $L(V_1, \cdots, V_n)$ with $V_1 \otimes V_2 \otimes \cdots V_n$, by declaring $\Psi(f) = \sum f(e_1^{i_1}, e_2^{i_2}, \cdots, e_n^{i_n}) e_1 \otimes \cdots e_n$ -- it's not hard to see this is a surjective mapping, and thus by dimensionality considerations, an isomorphism.
But in most algebra books, it's identified with $V_1^* \otimes V_2^* \otimes \cdots \otimes V^*_n$, by declaring the mapping $\Psi(\omega_1 \otimes \cdots \otimes \omega_n)(v_1, \cdots, v_n) = \omega_1(v_1)\omega_2(v_2)\cdots \omega_n(v_n)$, and then using linearity. This map is easily seen to be injective, and thus an isomoprhism.
Why are we focusing on the isomoprhism $L(V_1, \cdots, V_n) \cong V_1^* \otimes V_2^* \otimes \cdots \otimes V^*_n$ rather than the one $L(V_1, \cdots, V_n) \cong V_1 \otimes \cdots \otimes V_n$ ?
For $n=1$ you have $L(V_1) = V_1^*$ by definition of dual vector space, so you can naturally generalise this for $n>1$. In case you have inner products on each $V_i$ then $V_i \cong V_i^*$ canonically, so the two identifications are essentially the same.