Yes, that doesn't make much of a differerence. For the finite-dimensional case the theorem is proved by induction (just in each step, take one element of a complementary subspace and expand the functional on it), and for the infinite dimensional as well, with the additional help of Zorn's lemma.

\Edit: Maybe its clearest if one realizes that multilinear functionals of n variables are nothing but linear functionals on the n-times tensor product of V. That way you can just apply the usual Hahn-Banach theorem.