Writing a paper on algebraic surfaces, I was led to consider the finite group $\mathsf{H}(A)$ whose presentation is the following.
I start with an anti-symmetric matrix $A=(a_{ij})$ of order $2n$ over $\mathbb{Z}_p$ (the finite group/field with $p$ elements), $p \geq 3$, and I consider the following system of generators and relations:
Generators:
$x_1, \ldots, x_{2n}, \; z$
Relations: \begin{equation} \begin{split} x_1^p &= \ldots=x_{2n}^p=z^p=1 \\ [x_1, \, z] &= \ldots = [x_{2n}, \, z]=1\\ [x_i, \, x_j]& =z^{a_{ij}} \end{split} \end{equation} where, by slight abuse of notation, the exponent $a_{ij}$ stands for any lifting in $\mathbb{Z}$ of $a_{ij} \in \mathbb{Z_p}$.
I can prove the following
Proposition 1. If $\det A \neq 0$, then the group $\mathsf{H}(A)$ presented as above is extra-special of order $p^{2n+1}$ and exponent $p$.
Proof. Calling $\omega$ the symplectic form on the $\mathbb{Z}_p$-vector space $V:=(\mathbb{Z}_p)^{2n}$ whose matrix (with respect to the standard basis) is $A$, one checks that $\mathsf{H}(A)$ is isomorphic to the Heisenberg group $\mathsf{H}(V, \, \omega)$. By standard linear algebra, up to a change of coordinates the symplectic form $\omega$ can be transformed into the standard symplectic form $\omega_{\mathrm{st}}$, so our group is also isomorphic to $\mathsf{H}(V, \, \omega_{\mathrm{st}})$, that has the desired properties, being the central product of $n$ copies of the $p$-group of order $p^3$ and exponent $p$. $\square$
Now, I was asked by the referee to give a direct proof of Proposition 1, namely a proof not involving symplectic forms. The motivation for this request is that the present proof is a bit far from the spirit of the paper, whose emphasis is mainly on group presentations.
Of course, I can give an equivalent proof involving nonsingular anti-symmetric matrices instead of symplectic forms, but I suspect that this is not what he/she/other pronoun was asking for. So let me ask the following two questions.
Question 1. Is there a proof of Proposition 1 essentially different from the one I gave?
Question 2. Is there any precise reference for Proposition 1?
I suspect that all of this is very well-known to the experts. Any hint to orientate myself in the extensive literature on $p$-groups would be greatly appreciated.