Let $M$ be a $2n\times 2n$ symmetric matrix with homogeneous degree $1$ entries in the ring $k[x_1,\dots,x_n]$, where $k$ is an algebraically closed field. Assume that the determinant of $M$ is $f^2$, where $f$ is an irreducible, homogeneous polynomial of degree $n$.
Is it true that we can find a matrix $P$ such that $PMP^t$ is of the form $$\left(\begin{array}{c|c} 0 & N \\ \hline N^t & 0 \end{array}\right) $$ where $N$ is a $n\times n$ matrix with $\det(N)=\lambda f$, for $\lambda\in k^*$?