In what follows, I use "rotation" in a sense more closely mirroring that of everyday usage -- specifically I mean a smooth path between two elements of $SO(n)$ (since in everyday usage a rotation means progressing through all intermediate states between the initial and final states (see for example pp. 52-53 of Stillwell's Naive Lie Theory, which inspired this question, in part).
Without loss of generality, let the smooth path begin at $Id \in SO(n)$ and end at some arbitrary $C \in SO(n)$ -- this is because, given any smooth path $A \to B$, we can get a path $Id \to A^{-1} B$ by taking the left action of $A^{-1}$ which is a homeomorphism (and even a diffeomorphism, I think).
Question: Do the geodesics of $SO(n)$ correspond to our "intuitive sense" of rotations in $\mathbb{R}^3$, i.e. the original position and the final position being separated by rotation through a single plane, as opposed to the possibility of having to compose multiple rotations occurring in multiple planes?
Remember that I am only concerned with the geodesic path between two elements of $SO(n)$ here, the fact that the path is a geodesic meaning to imply that it is in some sense the "natural" path of intermediate states to rotate through in order to arrive at one position from another.
It would make some sense if this were the case, because then any such path would be an embedding of part of $SO(2)$ (the unit circle) into $SO(n)$, and $SO(2)$ is one-dimensional, thus an embedding of a compact subset of it into $SO(n)$ could conceivably be a path.
I am not quite sure how to phrase this question in a rigorous manner, although I have tried -- please let me know if anything is still unclear so that I can correct and improve the clarity of the question.
Geodesic is related with a metric Since $SO(n)$ is compact so there exists a biinvariant metric so that $$ \nabla_XY = \frac{1}{2} [X,Y]\ \ast$$ where $X,\ Y$ are left invariant vector field.
Note that $T_ISO(n)$ is set of skew symmetric matrices And define $$c_X(t):= \sum_{i=0}^\infty \frac{(tX)^i}{i!},\ X\in T_ISO(n)$$
Then $c_X'(t)=c_X(t) X$ so that $c_X$ is a integral curve at $I$ for a left invariant vector field $X$ Note that by $\ast$ $c_X$ is a geodesic