Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf\_0(Z):\dots:pf\_n(Z)]$ where the $pf\_i(Z)$ are the principal Pfaffians of order $n$ of $Z$. 


Is the closure of the image in $\mathbb{P}^n$ of this rational map a known variety? 



Rational map given by pfaffians
At least in characteristic zero, the image of this map is all of $\mathbb P^n$. In finite characteristic, I am not sufficiently familiar with Pfaffians to say whether the argument carries through.


Indeed, start by noting that the map is only well-defined for $Z$ of rank $n$, since otherwise all principal Pfaffians of rank $n$ vanish. For $Z$ of maximal rank, consider instead the map that sends $Z$ to $\left[(-1)^0pf\_0(Z):\dotsm:(-1)^npf\_n(Z)\right]$, which agrees with your map up to an automorphism of $\mathbb P^n$. This map sends $Z$ to its kernel, and every line arises as the kernel of some skew-symmetric matrix.


