Recall Schur's Pfaffian identity:
$$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$
Here $x_1,x_2\cdots x_{2n}$ are $2n$ variables, and $\mathrm{Pf}(A_{ij})$ stands for the Pfaffian of $2n$ by $2n$ skew-symmetric matrix $A$, whose $(i,j)$-th entry is $A_{ij}$.
Now, I would like to know whether there is similar closed form for the following Pfaffian(s):
$$ \mathrm{Pf}\left(\frac{x_j+x_i}{x_j-x_i}\right)_{1\le i,j\le 2n}, $$
and
$$ \mathrm{Pf}\left(\frac{1}{x_j-x_i}\right)_{1\le i,j\le 2n}. $$
By closed form, I mean the expression resembling the RHS of Schur's Pfaffian identity, i.e. only products are involved.
This question is motivated by (my) research in physics. The second Pfaffian is related to Majorana fermions and fractional quantum Hall states (the Moore-Read state). The first Pfaffian is a slightly modified form of the second one.
Thank you,
Isidore