How do I get from the universal product of the tensor product to other definitions.

Question

I was wondering how you can "derive" the common (or "classical") definition of the tensor product before the universal property was established (I think there is no need to repeat it here), i.e. a multilinear map of the cartesian product of $r$ copies of a vector space $V$ and $s$ copies of the dual space $V^*$ into, let's say, $\mathbb{R}$, from the universal defintion?