If $V$ is an $n$-dimensional real vector space then the $k$th exterior power $\Lambda^kV$ can be defined in a bunch of different ways and everyone seems to have their own preference. I've been exposed to a couple and I'm trying to verify for myself that they're all equivalent.
In particular, I've seen $\Lambda^kV$ defined as a quotient of $V^{\otimes k}$ by a subspace in two ways: One author uses the subspace spanned by elements of the form $v_1 \otimes \cdots \otimes v_k$ where $v_i = v_j$ for some $i \neq j$, and another author uses the subspace spanned by elements of the form $v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(k)} - (\operatorname{sgn}{\sigma})v_1\otimes \cdots \otimes v_k$ where $\sigma \in S_k$. Are these the same subspace? Or are they distinct subspaces such that the quotients by each can be identified in a natural way?