Singularities of Pfaffian hypersurfaces

As Sasha proved the general Pfaffian is smooth. On the other hand the special Pfaffian $X$ you wrote is an irreducible hypersurface of degree $6$ in $\mathbb{P}^4$ singular along a smooth curve $C$ of degree $20$ and genus $26$. Indeed $X$ has ordinary double points along $C$.

Your Pfaffian is indeed birational to the moduli space of $(1,11)$-polarized abelian surfaces, endowed with a symmetric theta structure and an odd theta characteristic. In this sense you can find a detailed description of this hypersurface in Lemma $2.1$ of this paper:

M. GROSS, S. POPESCU, The Moduli Space of (1,11)-Polarized Abelian Surfaces is Unirational, Compositio Mathematica 126: 1-23, 2001.