So, if we are in $\Bbb R^{n}$ there are $n(n-1)/2$ rotations. Which is to say:
Given $1$ of the $n$ axis, we can rotate it onto $n-1$ other axis. We then divide by $2$ to not overcount.
That's OK, however if we are in a 2D plane, rotating $x$ axis onto $y$ axis means rotating the plane around a $z$ axis which is outside the 2D space we are considering and inside $\Bbb R^{3}$.
If we instead rotate one of the 3 axis in $\Bbb R^{3}$, all the rotations are around axis inside the space we are considering.
Why this? Does this happen even in higher dimension? Is it a quirk of 3D spaces and a few others, or is it instead a quirk of 2D space?
No it doesn't. That's one way to think about the rotation, especially since we are so used to working in $3$ dimensions physically ourselves (and thinking about e.g. wheels and axles), but you don't actually have to talk about this third axis if you don't want to. A rotation in $\mathbb{R}^2$ is simply a rotation around the origin, and that's it.