Compact open topology on $\operatorname{GL}(n, \mathbb{R})$ coincides with Euclidean topology.

The universal property of the compact-open topology ("mapping space") as in your link means nothing else than that for locally compact $Y$

(1) the evalation map $e : Z^Y \times Y \to Z, e(f,y) = f(x)$ is continuous

(2) the exponential correspondence $E : Z^{X \times Y} \to (Z^Y)^X$ is a bijection

To prove that $i : GL(n, \mathbb{R}) \to Maps(\mathbb{R}^n, \mathbb{R}^n)$ is continuous, it therefore suffices to show that $\alpha = E^{-1}(i) : GL(n, \mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}^n$ is continuous. $\alpha$ is the restriction of the bilinear map $\tilde{\alpha} : End(\mathbb{R}^n) \times \mathbb{R}^n \to \mathbb{R}^n, \tilde{\alpha}(\phi,x) = \phi(x)$, where $End(\mathbb{R}^n)$ denotes the vector space of all endomorphisms of $\mathbb{R}^n$. But bilinear maps are continuous with respect to the Euclidean topologies (all occurring vector spaces are finite-dimensional).