Hello

This question somehow is related to a previous question I asked here. Let me set some notations. Let $\mathbb{F}_p$ be a finite field with $p$ element. Consider the following symmetric matrix

\begin{equation} J:=\begin{pmatrix} 0 & I_n \\\\ I_n & 0 \end{pmatrix}, \end{equation}

where $I_n$ is the identity matrix. Now the special orthogonal group is defined by

$$SO_{2n}(\mathbb{F}_p):=\{A\in SL_n(\mathbb{F}_p): AJA^T=J \}.$$

Obviously, when $\sigma,\tau\in M_n(\mathbb{F}_p)$ are skew-symmetric the following matrices belongs to the special orthogonal group. \begin{equation} \begin{pmatrix} I_n & \sigma\\\\ 0 & I_n \end{pmatrix}, \begin{pmatrix} I_n & 0\\\\ \tau & I_n \end{pmatrix}\in SO_{2n}(\mathbb{F}_p) \end{equation}

I have done some computations to show that these matrices generate $SO_{2n}(\mathbb{F}_p)$. But now, based on the question I have asked, I might have done some mistakes in my computations. Is it true that matrices mentioned above are a generating set?