A question about the symbol $n$ in the additive group $\mathbb{R}^n$

Question

As we know the symbol $n$ determines the dimension of $\mathbb{R}^n$ in the category of vector spaces. Also, in the category of manifolds, $n$ determines the dimension of $n$-manifold $\mathbb{R}^n$.

Now, my question is that:

Is there any algebraic property related to the additive group $(\mathbb{R}^n ,+)$ which is determined by $n$?

Answer 1

This answer proves that as additive groups $\mathbb{R}^n \cong \mathbb{R}^m$ for all $n,m$.