# Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries

I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.

I am really only interested in the ideal generated by submaximal Pfaffians of these matrices so in particular I am looking for particularly nice orbit representatives under multiplication by $SL (2n+1)$. For example, we may always assume that there are at most $3$ nonzero entries per row/column just by counting dimension. Similarly, we may also assume that the first row/column has at most 2 nonzero entries by a counting argument.

Just using the above guidelines, it is straightforward but tedious to do by hand the possibilities for low order matrices. For the general case, I am not sure how difficult the problem is.

As an example of what I mean by "nice", in the $5 \times 5$ case, I can assume every such skew symmetric matrix with linear forms $a_i$, $b_i$, $c_i$ as entries has Pfaffians obtained from a matrix of the form $$\begin{pmatrix} 0 & 0 & 0 & a_1 & a_2 \\ 0 & 0 & b_1 & b_2 & b_3 \\ 0 & -b_1 & 0 & c_1 & 0 \\ -a_1 & -b_2 & -c_1 & 0 & 0 \\ -a_2 & -b_3 & 0 & 0 & 0 \\ \end{pmatrix}$$ (with sufficient "genericness" conditions on the linear forms to ensure maximal rank).