  Let $\Theta_n = (\theta_1, \dots, \theta_n)$ and $\Xi_n = (\xi_1, \dots,
\xi_n)$ be two lists of $n$ variables and consider the diagonal action of
$\mathfrak{S}_n$ on the exterior algebra $\wedge \{ \Theta_n, \Xi_n \}$
generated by these variables. Jongwon Kim and the second author defined and
studied the fermionic diagonal coinvariant ring $FDR_n$ obtained from $\wedge
\{ \Theta_n, \Xi_n \}$ by modding out by the $\mathfrak{S}_n$-invariants with
vanishing constant term. On the other hand, the second author described an
action of $\mathfrak{S}_n$ on the vector space with basis given by noncrossing
set partitions of $\{1,\dots,n\}$ using a novel family of skein relations which
resolve crossings in set partitions. We give an isomorphism between a natural
Catalan-dimensional submodule of $FDR_n$ and the skein representation. To do
this, we show that set partition skein relations arise naturally in the context
of exterior algebras. Our approach yields an $\mathfrak{S}_n$-equivariant way
to resolve crossings in set partitions. We use fermions to clarify, sharpen,
and extend the theory of set partition crossing resolution.
