Let $V$ be a $k$-vector space ($k$ a field of arbitrary characteristic, or even just a commutative ring) and let $\alpha \colon V^r \to k$ and $\beta \colon V^s \to k$ be alternating multilinear forms on $V$. Define the wedge product $\alpha \wedge \beta \colon V^{r+s} \to k$ via the formula $$(\alpha \wedge \beta)(v_1, \dots, v_{r+s}) = \sum_{\sigma \in \mathrm{Sh}(r,s)} \mathrm{sign}(\sigma) \alpha(v_{\sigma(1)}, \dots, v_{\sigma(r)})\beta(v_{\sigma(r+1)}, \dots, v_{\sigma(r+s)}),$$ where $\mathrm{Sh(r,s)}$ denotes the set of shuffles where a shuffle is a permutation $\sigma$ of $\{1, \dots, r+s\}$ such that $\sigma(1) < \dots < \sigma(r)$ and $\sigma(r+1) < \dots < \sigma(r+s)$. This definition is given for example in Wikipedia.
I have a hard time showing that $\alpha \wedge \beta$ is again alternating. I cannot manage to modify the proof given for example here in a comment, where the sum is taken over all permutations of $\{1, \dots, s\}$ (which turns out to be equivalent in fields of characteristic $0$).