orbit representatives for the group of unipotent matrix acting on the set of skew-symmetric matrices

Question

Suppose $F$ is a p-adic field, I was trying to compute orbit representatives for the group of n by n upper-triangular unipotent matrces $U_n(F)$ acting on the set of n by n skew-symmetric matrices. I think there is some structure theorem of skew-symmetric matrices saying that any skew-symmetric matrix with coefficients in a field $F$ is of the form $gS^tg$ for some g in $M_n(F)$ and $S=\begin{bmatrix} \ \ & I_d \\ -I_d & \ \ \end{bmatrix}$ for $n=2d$, even. and $S=\begin{bmatrix} \ \ &\ \ & I_d \\ \ \ & 0 & \ \ \\ -I_d & \ \ & \ \ \\ \end{bmatrix}$ when $n=2d+1$ is odd. So the orbit representatives should corresponds to the ones of the action of $U_n(F)$ on $M_n(F)/Stab(S)$. But first I couldn't find a good way to think about the stablizer $Stab(S)$, and finding the orbit representatives are also not that easy. Does anyone have a good way to think about this? Thanks a lot. (For my concern it suffices to do this problem for some open dense subset of the set of skew-symmetric matrices $Sk_n(F)$ under the p-adic topolgy, I was thinking about to use $GL_n(F)\subset M_n(F)$ and use the Iwasawa decomposition $G=UAK$ where $G=GL_n(F)$, $U=U_n(F)$, and $K$ a maximal compact subgroup such that the decomposition holds, but I don't know wether this would help or not)