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.