Determining the axis of rotation of a special orthogonal matrix My syllabus states the following procedure to determine quickly the rotational axis of a 3x3 matrix $A$ that is orthogonal with determinant 1, but it's not completely clear:
"If $A$ $\in$ SO(3) and $A$ is not symmetric and $u$ is an eigenvector with eigenvalue 1 (so $u$ determines the axis of rotation), then construct the skew-symmetric part of $A$:
$$S = \frac{1}{2}(A-A^T) = \begin{bmatrix} 0 & \omega_3 &-\omega_2 \\
-\omega_3 & 0 & \omega_1 \\  \omega_2 & -\omega_1 & 0 
\end{bmatrix}$$
From $Au = u$, it follows that $Su = 0$, so the axis of rotation is determined by the vector $[\omega_1,\omega_2,\omega_3]^T$.
If $A$ is symmetric then $S = 0$. $A$ is in that case the identity matrix (in which case this theorem is ofcourse true), or $A$ is orthogonal equivalent with $diag(1,-1,-1)$ (in which case the theorem doesn't hold)."
I understand that if $u$ determines the axis of rotation, then $u$ is in the kernel of $S$, but I don't understand why it follows that $[\omega_1,\omega_2,\omega_3]^T$ (which is in the kernel of $S$) has to determine the axis of rotation, which seems to imply that the kernel of $S$ is 1-dimensional (if this is true, then the problem is solved, because the 1-dimensional space of vectors that determine the axis of rotation are already in the kernel, so there can be no other vectors in the kernel of $S$).
The second thing I don't understand is why $A$, if symmetric and not $I$, has to be orthogonal equivalent with $diag(1,-1,-1)$. 
I hope someone can help me out.

For your first question, $\ker S$ is indeed one-dimensional if $S$ is nonzero. You may verify that if $\omega_i\neq0$, then the two rows of $S$ containing $\omega_i$ are linearly independent.
For the second question, every real symmetric matrix $A$ is orthogonally equivalent to a real diagonal matrix that contains the real eigenvalues of $A$ as diagonal entries. Now $A$ is also real orthogonal, so these eigenvalues have to be $\pm1$. Since $\det A=1$ and $A\neq I$, the only possibility is that two of these eigenvalues are $-1$ and the remaining one is $1$.
