Let $\Lambda^kV^*$ be the space of alternating k-linear forms on $V^k:=V \times \dots \times V$.
Such that for $\omega \in \Lambda^kV^*$,
$$\omega(v_1, \dots, v_i+v_j,\dots, v_k)=\omega(v_1, \dots, v_i,\dots, v_k)+\omega(v_1, \dots,v_j,\dots, v_k), i,j\in \{1,\dots,k\}$$ $$\omega(v_{\pi(1)}, \dots,v_{\pi(k)})=sgn(\pi)\omega(v_1, \dots, v_k), \pi \in S_k.$$
The alternating operator $Alt^k$ is defined on $\bigotimes^kV^*$, the space of k-linear forms, so $\Lambda^kV^*\subset \bigotimes^kV^*$. $$Alt^k:\bigotimes^kV^* \longrightarrow \Lambda^kV^* $$ $$\omega(v_1,\dots,v_k) \longmapsto \frac{1}{k!}\sum_\limits{\sigma \in S_k}sgn(\sigma)\omega(v_{\sigma(1)}, \dots,v_{\sigma(k)})$$
So now I am trying to show for $\varphi \in \bigotimes^kV^*$ and $\psi \in \bigotimes^lV^*$ that:
$$Alt^{k+l}(\varphi \otimes\psi)=Alt^{k+l}(Alt^k(\varphi)\otimes \psi)$$ with, $\varphi \otimes\psi(v_1,\dots,v_{k+l})=\varphi(v_1,\dots,v_k)\psi(v_{k+1},\dots,v_{k+l})$.
My attempt: So I just tried to simply the right-hand of the equation, doing that we define: $$\tilde{\varphi}(v_1,\dots,v_k):=\frac{1}{k!}\sum_\limits{\mu \in S_k}sgn(\mu)\varphi(v_{\mu(1)}, \dots,v_{\mu(k)})=Alt^k(\varphi(v_1,\dots,v_k))$$ So that:
\begin{align*} Alt^{k+l}(Alt^k(\varphi)\otimes \psi)(v_1,\dots,v_{k+l}) =Alt^{k+l}(\tilde{\varphi}\otimes\psi)(v_1,\dots,v_{k+l})=\\ \frac{1}{(k+l)!}\sum_\limits{\sigma \in S_{k+l}}sgn(\sigma)\tilde{\varphi}(v_{\sigma(1)}, \dots,v_{\sigma(k)})\psi(v_{\sigma(k+1)}, \dots,v_{\sigma(k+l)})= \frac{1}{(k+l)!}\sum_\limits{\sigma \in S_{k+l}}sgn(\sigma)\left(\frac{1}{k!}\sum_\limits{\mu \in S_k}sgn(\mu)\varphi(v_{\mu(1)}, \dots,v_{\mu(k)})\right)\psi(v_{\sigma(k+1)}, \dots,v_{\sigma(k+l)})=\dots=\\ \frac{1}{(k+l)!}\sum_\limits{\sigma \in S_{k+l}}sgn(\sigma)\left(\frac{1}{k!}\sum_\limits{\mu \in S_k}\varphi(v_{\sigma(1)}, \dots,v_{\sigma(k)})\right)\psi(v_{\sigma(k+1)}, \dots,v_{\sigma(k+l)})=\\ \frac{1}{(k+l)!}\sum_\limits{\sigma \in S_{k+l}}sgn(\sigma)\frac{k!}{k!}\varphi(v_{\sigma(1)}, \dots,v_{\sigma(k)})\psi(v_{\sigma(k+1)}, \dots,v_{\sigma(k+l)})=Alt^{k+l}(\varphi \otimes\psi) \end{align*}
Question: I think that is the idea behind it, but I don't really now what to do, to get the equation, after the dots. I thought about using the the fact that $\tilde{\varphi}$ is alternating but, if we look at it with $\sigma \in S_{k+l}$:
\begin{align*} \frac{1}{k!}\sum_\limits{\mu \in S_k}sgn(\mu)\varphi(v_{\mu\circ\sigma(1)}, \dots,v_{\mu\circ\sigma(k)})=\tilde{\varphi}(v_{\sigma(1)}, \dots,v_{\sigma(k)})\neq sgn(\sigma)\tilde{\varphi}(v_{1}, \dots,v_{k}) \end{align*} ,since $\sigma \notin S_k$.
I would love to get a hint or idea (maybe how to approach the problem differently)!