I have done some research on vector alignment here on math exchange and in some technical papers. I seem to find some discrepancies in the formulae presented and I am not sure if these are errors or if they present slightly different concepts.
So the question in general is vector alignment via the Rodriguez Formula. The mathexchange post in question is this one mathexchange
The formula for R (from the post) is as follows:
$\mathit{R} = \mathit{I} + [v]_x + [v]_x^2\frac{1-c}{s^2}$
where $s = \sin(\theta)$ and $c = \cos(\theta)$
On wikipedia the formulae is as follows:
$\mathit{R} = \mathit{I} + s[v]_x + [v]_x^2(1-c)$
Using the notes of Ethan Eade we get:
$\mathit{R} = \mathit{I} + \frac{s}{\theta}[v]_x + [v]_x^2\frac{1-c}{\theta^2}$
Equation 82 on page 10.
So which one is correct? Or do they express slightly different things? I am inclined to believe Ethan Eade's notes as he has a track record of publications.
In Eade's article, the vector $\omega$ (which you've changed to $v$) is arbitrary, whereas in the wikipedia article, $\mathbf k$ (your $v$) is a unit vector. If you define $\theta = \|\omega\|$ (as Eade does in formula (10)), then $\omega = \theta\mathbf k$, and $[\omega]_\times = \theta \mathbf K$,so his formula now matches the one in wikipedia (see also this derivation).
Similarly, in the other Math Exchange thread, $v = a \times b$ is also not a unit vector. It's norm is $\sin \theta$. Thus $[v]_\times = \sin \theta \mathbf K$, and with this we see why that version of the equation drops the $s$ from the $[v]_\times$ term, and divides by $s^2$ in the $[v]_\times^2$ term.