Rotation matrices and exponential map
I'm having some trouble with rotation matrices. Basically, I want to show that any matrix , specified by an angle and a unit vector can be written as
where is a 'basis' matrix for , the space of all skew-symmetric 3x3 matrices. Note that I'm representing a rotation as a vector : the unit vector described the rotation, the length of the vector the amount of degrees
I know the exponential map is surjective, and that coincides with the rotation matrix representing the rotation around the standard basis vector with angle .
I see that surjectivity of exp implies that any R can be written as for some skew-symmetric matrix X. Given that the span , I get
This is looking like what I want to prove, but I'm kind of stuck here. Can anyone help?