Multilinear Mapping from Free Modules is Determined by Bases

Theorem
Let $R$ be a commutative ring with unity.

Let $M_1,\ldots,M_n$ be free $R$-modules.

Let $B_1,\ldots,B_n$ be bases of $M_1,\ldots,M_n$.

Let $N$ be an $R$-module.

Let $f:B_1\times\cdots\times B_n\to N$ be a function.

Then there exists a unique multilinear map $\phi:M_1\times\cdots\times M_n\to N$ such that $\phi(b)=f(b)$ for all $b\in B_1\times\cdots\times B_n$.

Also see

 * Universal Property of Free Modules