I am working with the definition of complexification $V_\mathbb{C}= V \otimes_\mathbb{R} \mathbb{C}$ where V is a real vector space, in order to better understand forms and vectors of type (p,q) when $V$ is itself endowed with an (almost) complex structure $J:V\rightarrow V$.
I am trying understand formulas like $$ \Lambda ^k V_\mathbb{C} = \left(\Lambda^k V\right) \otimes \mathbb{C} $$ but in order to work with wedge products I first need to convince myself that $$ (V\otimes_\mathbb{R} \mathbb{C}) \otimes_\mathbb{C} (V\otimes_\mathbb{R} \mathbb{C}) \simeq (V\otimes_\mathbb{R} V)\otimes_\mathbb{R} \mathbb{C} $$ I am trying to come up with a proof interpreting 2-tensors as bilinear maps but I can't seem to find a nice and simple formula for the isomorphism. Any ideas?