differential geometry - Coset space and transitiviy

I have a question regarding coset space or homogeneous space $SO(n+1)/SO(n)$ which is simply $S^n$. I need some intuition regarding this result.

As everyone knows that for a simple case of $SO(3)/SO(2)$, one can have $SO(3)$ as a group acting on $\mathbb{R}^3$ and $SO(2)$ as an isotropy group of $x\in\mathbb{R}^3$, then the group $SO(3)$ acts transitively on $S^2$ and we get $S^2$ as the coset.

Since the result is just 2-sphere or $n$-sphere, is there an intuitive way of seeing it?