How to prove the decomposition of k-forms over a complexified vector space?

Question

I am studying the book "Complex Geometry, an introduction", page 27. We consider the complexification of a vector space V, namely $V_\mathbb{C}$. If we take the decomposition $$V = W_1 + W_2.$$ Then the following decomposition of the space of k-forms over $V_\mathbb{C}$ follows: \begin{equation} \bigwedge^{k} V_\mathbb{C} = \bigoplus_{p+q = k} \bigwedge^p W_1 \otimes_\mathbb{C} \bigwedge^q W_2 \end{equation} But I do not know how to prove it. Even understanding the equalities as isomorphisms, if $\alpha \otimes \beta \in \bigwedge^p W_1 \otimes_\mathbb{C} \bigwedge^q W_2 $. How do I define an element of $\bigwedge^{k} V_\mathbb{C}$ from $\alpha \otimes \beta$ ? My biggest problem is that $\alpha$ only p vectors from $W_1$ and $\beta$ only q vectors from $W_2$, but I need to construct an (antisymmetric) object that takes p+q vectors from $V_\mathbb{C}$.