Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?



Parametrization of O(3)
The general element is $\pm\exp(A)$ where $A$ is skew-symmetric.
(This gives each element infinitely often). This trick essentially
works for all compact Lie groups.


There is also the Cayley parameterization: $(I+A)(I-A)^{-1}$
for skew-symmetric $A$
is the general element of $SO(3)$ which lacks an eigenvalue $-1$
(so isn't a half-turn.) This parameterizes all such matrices
once each.


