Picture behind $SO(3)/SO(2)\simeq S^2$ Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$

(To remove this question from the unanswered list):

> Think of group action. $SO(3)$ acts on $\mathbb S^2$, and $SO(2)$ is a stabilizer (any $SO(3)$ element fixing $(0,0,1)$ must be the rotation of the $x,y$ plane)