Explicit computation of $H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z})$.

Question

I'm interested in the computation of the second cohomology group of the elementary abelian group $\mathbb{F}_p^n$ with coefficients in $\mathbb{R}/\mathbb{Z}$:

$$H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z}) \cong \mathbb{F}_p^{n(n-1)/2}$$

where for simplicity let's suppose $p$ is odd. I'm pretty sure this is right.

The following discussion of how the right hand side arises is cribbed from [Tata Lectures on Theta III, Mumford, Nori and Norman]; any mistakes my own.

The space $\mathbb{F}_p^{n(n-1)/2}$ on the right corresponds naturally to the space of alternating bilinear maps $\mathbb{F}_p^n \times \mathbb{F}_p^n \to \mathbb{R}/\mathbb{Z}$. The link to $H^2$ comes from looking at the commutator map $G \times G \to \mathbb{R}/\mathbb{Z}$ in the corresponding central extension

$$0 \to \mathbb{R}/\mathbb{Z} \to G \to \mathbb{F}_p^n \to 0$$

and noting it descends to an alternating bilinear map $\mathbb{F}_p^n \times \mathbb{F}_p^n \to \mathbb{R}/\mathbb{Z}$ on the quotient. If $\rho \colon \mathbb{F}_p^n \times \mathbb{F}_p^n \to \mathbb{R}/\mathbb{Z}$ is an explicit 2-cocycle, then this alternating form is just $$\sigma(a,b) = \rho(a,b) - \rho(b,a) $$ and is readily seen to be invariant under adding coboundaries to $\rho$.

So, this defines a homomorphism $$H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z}) \to \mathbb{F}_p^{n(n-1)/2}$$ and it's not hard to show it's surjective, i.e., to construct very explicit Heisenberg-group-type examples of group extensions or cocycles showing that every $\sigma$ arises in this way. It is also clear that the kernel consists precisely of even cocycles (i.e., $\rho(a,b) = \rho(b,a)$) modulo coboundaries.

My question is:

Question: can someone give me an fairly explicit proof that this map is injective, and therefore that $H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z})$ is what we think it is?

At some point a proof must use the fact that $\mathbb{R}/\mathbb{Z}$ hasn't been replaced by, say, $\mathbb{F}_p$, as then cocycles corresponding to $\mathbb{Z}/p^2 \mathbb{Z}$ will be in the kernel.

What I would love to have, in order of preference, is:

1. an algorithm that takes any even cocycle on $\mathbb{F}_p^n$ and spits out an explicit function whose coboundary is $\rho$;

2. any proof from which such an algorithm could in principle be recovered, even if doing so involves a lot of work diving into proofs of standard results; or

3. any proof at all.

As ever, any pointers or references greatly appreciated.