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''of a skew-symmetric matrix $X$''

The polynomial  $\text{Pf } X$ in the entries of $X$ whose square is $\text{det } X$.  More precisely, if $X = \|x_{ij}\|$ is a skew-symmetric matrix (i.e.  $x_{ij}=-x_{ji}$, $x_{ii}=0$; such a matrix is sometimes also called an  alternating matrix) of order $2n$ over a commutative-associative ring  $A$ with a unit, then $\text{Pf } X$ is the element of $A$ given by the  formula

$$
\text{Pf } X = \sum_s \varepsilon(s)x_{i_1j_1}\ldots x_{i_nj_n},
$$

where the  summation is over all possible partitions <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250013.png" /> of the  set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250014.png" /> into  non-intersecting pairs <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250015.png" />, where one may  suppose that <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250016.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250017.png" />, and where  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250018.png" /> is the sign of  the permutation

<table class="eq"  style="width:100%;"> <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250019.png"  /></td> </tr></table>

A Pfaffian has the following properties:

1)  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250020.png" /> for any matrix  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250021.png" /> of order <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250022.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250023.png" />;

3)  if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250024.png" /> is a free <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250025.png" />-module with basis  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250026.png" /> and if

<table  class="eq" style="width:100%;"> <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250027.png"  /></td> </tr></table>

then

<table  class="eq" style="width:100%;"> <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250028.png"  /></td> </tr></table>

====References====
<table><TR><TD  valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,    "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' ,  Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from  French)</TD></TR></table>