Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$.
When $n=2$, it is known that $M$ is just the 2-sphere.
1) On the 2-sphere, the exponential map at any point is surjective, is that true?
2) If yes, is this fact true for all $n$ ? In other words, is it true that for all $n$, the exponential map at any point of $M$ is surjective onto $M$ ?