I need to calculate some tensor products and stuck the whole night at this simple things. But finally, I confused myself.
When I have a commutative ring $R=\mathbb{C}[x_1,...,x_n]$ and considering the enveloping algebra of it $R^e=R \otimes_{\mathbb{C}} R^{op}$, then $R^e \cong R^2$ since $R$ is commutative. A $R$-bimodul $M$, is a left $R^e$-module. This means that esspecially $M=R$ is a left $R^e$-module. So in order we have $R^e \otimes_{R^e} M \cong M$. I hope it is correct unit here. This, in turn, implies that
$(R^e)^m \otimes_{R^e} R \cong R^{2m} \otimes_{R^2} R \cong R^{m+1}$.
But what is about $\Lambda^2((R^e)^m) \otimes_{R^e} R$ ?