Group Actions on $\Bbb R^n$

Question

We are learning about Quotient spaces, and group actions. For any $n\in\mathbb{N}$. We know that the function $G(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=1\}$ is a group action on $\mathbb{R}^n$. I am struggling to see what the actual orbits are, and why $\mathbb{R}^n\backslash G(n)$ is homeomorphic to the positive real line. These are given as fact, but I dont see why these are elementary results.