## Rotation matrices and exponential map

Hi guys,

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

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

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

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

$\displaystyle R= exp (\sum a_i J_i)$.

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