Hi guys,

I'm having some trouble with rotation matrices. Basically, I want to show that any matrix R(n \alpha) \in SO(3), specified by an angle \alpha and a unit vector n\in S^2 can be written as

R(n \alpha) = exp (\alpha \sum n_i J_i ),

where J_i is a 'basis' matrix for \mathfrak{so}(3), 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 exp : \mathfrak{so}(3) \to SO(3) is surjective, and that exp[\alpha J_i] coincides with the rotation matrix representing the rotation around the standard basis vector e_i with angle \alpha.

I see that surjectivity of exp implies that any R can be written as R = exp (X) for some skew-symmetric matrix X. Given that the J_i span \mathfrak{so}(3), I get

R= exp (\sum a_i J_i).

This is looking like what I want to prove, but I'm kind of stuck here. Can anyone help?