Let
S
S
be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let
I
I
be the determinantal ideal of maximal minors or
P
f
Pf
the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the
S
S
-module structures of
E
x
t
S
j
(
S
/
I
t
,
S
)
Ext^j_S(S/I^t, S)
and
E
x
t
S
j
(
S
/
P
f
t
,
S
)
Ext^j_S(S/Pf^t, S)
, from which we get the degrees of generators of these
E
x
t
Ext
modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form
H
m
j
(
S
/
I
t
)
H^j_\mathfrak {m}(S/I^t)
.