Suppose we are working over a field $k$. Consider the affine space $k^n$ and the $\mathbb{Z}/(2)$-action on it given by $(x_1,x_2,\dots,x_n) \mapsto (-x_1,-x_2,\dots,-x_n)$. I would like to compute the coordinate ring of the categorical quotient $k^n//(\mathbb{Z}/(2))$ under this action. This coordinate ring is exactly the invariant ring $k[x_1,x_2,\dots,x_n]^{\mathbb{Z}/(2)}$, which I would like to express in terms of generators and relations.
Let $p_{ij}=x_ix_j$ for all $i,j$. Note that $p_{ij}=p_{ji}$. It is easy to see that the invariant ring is generated by polynomials of the form $p_{ij}=x_ix_j$ for $i\leq j$. Among these generators, we have the relations $p_{ij}p_{kl} = p_{il}p_{jk}$. I would like to show that these are all, that is, the ideal of all relations is generated by the elements $R_{ijkl}=p_{ij}p_{kl} - p_{il}p_{jk}$.
It's easy to see this for small $n$, and I think I can write a messy argument to show that any relation can be expressed in terms of these, but I was wondering if there is a nice slick way to see this. I would also not be surprised if this is something standard, in which case I would quite appreciate a reference!