Spivak defines the wedge product as
$\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$
and I have been running into some conceptual issues here. The alternation is defined as
$\text{Alt}(T)(v_1, \dots , v_n) = \frac{1}{k!}\sum_{\sigma \in S} \text{sgn}(\sigma) \cdot T(v_{\sigma(1)}, \dots , v_{\sigma(n)})$
which seems to me to be some type of weighted average of the possible values of T (which is multilinear). I am just failing to see how this plays into the definition of the wedge product. Reading some other books, it seems like the wedge product helps to form operators from a k-dimensional space to $R$. I'm just not seeing the motivation for this specific definition of the product, and it is really making it difficult to understand differential forms.
Any advice here or in general?