Let as usual $GL_n$ be the scheme given by the equation $$det(\{x_{kl}\}_{1\leq k,l\leq n})y=1$$ in $\mathbb{A}^{{n^2}+1}$. I have seen that one considers elements of $GL_n$ likewise as linear isomorphisms $\mathbb{A}^n\to\mathbb{A}^n$ of schemes. I want to understand what this means formally and it would be nice if something like the following holds:
Perhaps the scheme $GL_n$ represents some functor of points $L:Sch^{op}\to Sets$ which in turn can easily be identified with linear isomorphisms $\mathbb{A}^n\to\mathbb{A}^n$. In other words, there is perhaps an isomorphism $$ Hom_{Sch}(-,GL_n)\cong L(-) $$ of functors from $Sch^{op}$ to $Sets$.
What could this functor $L$ be? Perhaps $L(V)$ is the set of linear isomorphisms $\mathbb{A}^n\times V\to\mathbb{A}^n\times V$ of schemes? This is a functor from $Sch^{op}$ to $Sets$ by pulling back along $\mathbb{A}^n\times f$ for a morphism $f:W\to V$ in $Sch$ but this is just a guess for $L$.
My question is:
Is there a way to understand $GL_n$ in these terms as the representing object of a certain functor $L(-)$ which can easily be identified with the linear isomorphisms $\mathbb{A}^n\to\mathbb{A}^n$?
I know that an $R$-point of $GL_n$ can be viewed as an invertible $(n\times n)$-matrix $\{a_{kl}\}_{1\leq k,l\leq n}$ with entries in $R$. This defines a linear morphism $\mathbb{A}^n\times V\to\mathbb{A}^n\times V$ where $V=\operatorname{Spec}(R)$ through an automomorphism on $R[x_1,\ldots,x_n]$ sending $x_k$ to $\sum_{l=0}^n a_{kl} x_l$ (I hope this is correct). The question is, how to express this fact precisely in the language above.
This is standard and can be found in any book on algebraic groups. The scheme $\mathrm{GL}_n$ represents the contravariant functor $X \mapsto \mathrm{GL}_n(\Gamma(X,\mathcal{O}_X))$ (actually this is the functorial definition of $\mathrm{GL}_n$). For a commutative ring $R$, the group $\mathrm{GL}_n(R)$ is isomorphic to the group of linear automorphisms of $\mathbb{A}^n_R$.
If $X$ is an arbitrary scheme, then one has to be careful as for the definition of linear. In topology one usually demands this fiberwise, but this does not suffice here when $X$ is not reduced. A morphism $\mathbb{A}^n_X \to \mathbb{A}^m_X$ is called linear if for every $X$-scheme $T$ the induced map on $T$-valued points $\Gamma(T,\mathcal{O}_T)^n \to \Gamma(T,\mathcal{O}_T)^m$ is $\Gamma(T,\mathcal{O}_T)$-linear. Then it is induced by a matrix, and all these matrices have to be compatible. We see that the linear morphisms correspond to $M_{m \times n}(\Gamma(X,\mathcal{O}_X))$. In particular, the linear automorphisms of $\mathbb{A}^n_X$ correspond to $\mathrm{GL}_n(X)$.