I would like to show these two definitions of wedge product (of differential $k$ and $l$ forms are equivalent). The first definition is:
$$(\alpha \wedge \beta)(v_1, \cdots, v_{k+l}) = \frac{1}{k! l!} \sum_\sigma \text{sgn}(\sigma) \, \alpha(v_{\sigma(1)}, \cdots, v_{\sigma(k)}) \, \beta (v_{\sigma(k+1)}, \cdots, v_{\sigma(k+l)})$$
I.e this specifies the action of the wedge product on $k + l$ vectors. The other definition I have is this:
If we let $I$, $J$ denote $k$,$l$ multi-indices, and write:
$$ \alpha= \sum_{I}a_I dx_I, \qquad \beta = \sum_{J} b_J dx_J $$ Then the wedge product is given by:
$$ \alpha \wedge \beta = \sum_{IJ}a_I b_J dx_I \wedge dx_J $$
I am not sure where to begin proving this, mainly because one definition only specifies what occurs when one evaluates the form.