As a vector space, it is clear that the automorphism group of $\mathbb{R}^n$ is $\mathsf{GL}_n(\mathbb{R})$.
My question is: What is the $\text{Aut}((\mathbb{R}^n,+))$ ($n \ge 2$) as a group? Clearly, $\text{Aut}((\mathbb{R}^n,+))$ contains $\mathsf{GL}_n(\mathbb{R})$.
Given that the automorphism group of $\mathbb{R}$ is enormous, to simplify matters, I would greatly appreciate an answer with respect to continuous automorphisms (if this is known).
The question above is related to this question concerning additive maps that do not preserve scalar multiplication. In particular, are there continuous isomorphisms of $\mathbb{R}^n$ ($n \ge 2$) that do not preserve scalar multiplication? (Clearly, this would indicate that the containment above is strict.)