Let $\omega\in\Lambda^kV^*$ and $\eta\in\Lambda^lV^*$
Show that \begin{equation} \omega\wedge\eta=\frac{(k+l)!}{k!l!}Alt(\omega\otimes\eta) \end{equation}
What i know:
$(\omega\wedge\eta)(v_1,...,v_k,v_{k+1},...,v_{k+l}) =\frac{1}{k!l!}\sum_{\sigma\in S_{k+l}} sgn(\sigma)\omega(v_{\sigma(1)},...,v_{\sigma(k)}\eta(v_{\sigma(k+1)},...,v_{\sigma(k+l)}) $
and \begin{equation} (\omega\wedge\eta):=\sum_{i_1<...<i_k,j_1<...<j_l}a_{i_1,...,i_k}b_{j_1,...,j_l}(\psi_{i_1}\wedge ... \wedge\psi_{i_k}\wedge\psi_{j_1}\wedge...\wedge\psi_{j_l}) \end{equation} with coefficients $a_{i_1,...,i_k},b_{j_1,...,j_l}\in \mathbb{R}$ and $(\psi_{1},...,\psi_n)$basis of $V^*$
So how can i find the relation between these two defininitions? Any hint or help is appreciated.