For $n$ a positive integer, I would like to define $PGL_n$ as a group scheme. One candidate is $X:=\mathrm{Proj}\big(\mathbb{Z}[x_{ij}:1\leq i,j\leq n]\big)\,\backslash \,\mathbb{V}(det)$, but I am not sure if this is correct/reasonable. If $K$ is a field, $X(K)$ is identified with the set of invertible $n\times n$ matrices over $K$, modulo scaling by an element of $K^\times$.
What does $X(R)$ look like for an arbitrary commutative ring $R$?
If I am not mistaken, $X$ is affine with coordinate ring $A:=\mathbb{Z}[x_{ij}][det^{-1}]_0$ (the subscript $0$ means the homogeneous elements of degree $0$). So $X(R)$ should be the set of ring homomorphisms $A\to R$. It is difficult for me to understand this set of homomorphisms, and I'm hoping to see a description of $X(R)$ as some set of (equivalence classes of) matrices.