  Let $S$ be the coordinate ring of the space of $n\times n$ complex
skew-symmetric matrices. This ring has an action of the group
$\textrm{GL}_n(\mathbb{C})$ induced by the action on the space of matrices. For
every invariant ideal $I\subseteq S$, we provide an explicit description of the
modules $\textrm{Ext}^{\bullet}_S(S/I,S)$ in terms of irreducible
representations. This allows us to give formulas for the regularity of basic
invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to
characterize when these ideals have a linear free resolution. In addition,
given an inclusion of invariant ideals $I\supseteq J$, we compute the
(co)kernel of the induced map $\textrm{Ext}^j_S(S/I,S)\to
\textrm{Ext}^j_S(S/J,S)$ for all $j\geq 0$. As a consequence, we show that if
an invariant ideal $I$ is unmixed, then the induced maps
$\textrm{Ext}_S^j(S/I,S)\to H_I^j(S)$ are injective, answering a question of
Eisenbud-Musta\c{t}\u{a}-Stillman in the case of Pfaffian thickenings. Finally,
using our Ext computations and local duality, we verify an instance of Kodaira
vanishing in the sense described in the recent work of
Bhatt-Blickle-Lyubeznik-Singh-Zhang.
