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Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...