Is $SO(n)$ diffeomorphic to $SO(n-1) \times S^{n-1}$ There is a fibration $SO(n-1) \mapsto SO(n) \mapsto S^{n-1}$, from basically taking the first column of the matrix in $\mathbb{R}^n$. Is this fibration trivializable? 

If $n=3$, the Hopf fibration is the composition $S^3=Spin(3)\rightarrow SO(3)\rightarrow S^2$, so if $SO(3)\rightarrow S^2$ is trivial, so  the hopf fibration will be flat and this is not true.
https://en.wikipedia.org/wiki/Hopf_fibration#Geometric_interpretation_using_rotations
