Is there a continuous injective group homomorphism $GL_n(\mathbb{R}) \to O(n)$?
I'm struggling to construct one - given a matrix $A$ in $GL_n(\mathbb{R})$, how can we construct a matrix $O(n)$ in terms of $A$? I've tried constructions like $AA^T$, $A^{-1}A^T$, $A+A^T$, $A^{-1}+A^T$, and none work (not even mapping to $O(n)$, let alone being injective).
There is not any continuous injection $GL_n(\mathbb{R})\to O(n)$ (for $n>0$). This follows from the invariance of domain theorem, which implies in particular that a nonempty open subset of $\mathbb{R}^n$ has no continuous injection to $\mathbb{R}^m$ for $m<n$. Since $GL_n(\mathbb{R})$ is a manifold of dimension $n^2$ and $O(n)$ is a manifold of dimension $n(n-1)/2$, a continuous injection $GL_n(\mathbb{R})\to O(n)$ would locally give a continuous injection from an open subset of $\mathbb{R}^{n^2}$ to $\mathbb{R}^{n(n-1)/2}$, which is impossible.