Let $M$ be an $R$-module. Let $T^k_R(M)$ be the $k$-th tensor power of $M$ over $R$.
The $k$-th exterior power is defined as $T^k_R(M)$ modulo the subspace generated by all pure tensors $m_1 \otimes \cdots \otimes m_k$ such that $m_i=m_j$ for some $i \ne j$.
Let $W=\langle (m_1 \otimes \cdots \otimes m_k) + (m_{\tau(1)} \otimes \cdots \otimes m_{\tau(k)}) \mid \tau=(ij) \in S_k \text{ is a transposition} \rangle$.
Is the $k$-th exterior power (the set of alternating tensors) equal to $T^k_R(M)/W$?
I find that $T^k_R(M)/W$ satisfies the universal property for alternating $R$-multilinear maps: that is, if $f: M^k \to P$ is a $R$-multilinear and alternating, then $f$ induces a linear map $\bar f: T^k_R(M)/W \to P$.