I want to show, that the $\wedge$-product of multilinear forms is alternating. Therefore
$\omega\wedge\eta(\dotso, v,\dotso, v,\dotso)=0$
Since $\omega$ and $\eta$ are already alternating multilinear forms, there is only one interesting case. The case
$\frac{1}{r!s!}\sum_{\tau\in S_{r+s}} sgn(\tau)\omega(v_{\tau(1)},\dotso,v,\dotso,v_{\tau(r)})\eta(v_{\tau(r+1)},\dotso,v,\dotso,v_{\tau(r+s)})\stackrel{?}{=}0$
I do not see, how to get this result. Is there an easy way, or is it an exhausting calculation?
Thanks in advance.