First of all, terminology. SO(n) is orientation-preserving orthogonal transformations on n-space, or equivalently the orientation-preserving symmetries of an (n-1)-sphere in n-space. So Joshua's statement is about SO(n) for n>3.
OK. So the obvious way to interpret "rotation about an axis" in many dimensions is: you choose a 2-dimensional subspace V, then represent an arbitrary vector as v+w with v in V and w in its orthogonal complement, and then you rotate v. The dimension of the set of these things is (n-1)+(n-2) from choosing V -- you can pick one unit vector to be in V, and then another unit vector orthogonal to it -- plus 1 from choosing how far to rotate. So, 2n-2.
And yes, the dimension of SO(n) is n(n-1)/2. One way to see this: you've got matrices with n^2 elements, and n(n+1)/2 constraints on those elements because all the pairwise inner products of the columns (including each column with itself) are specified.
These dimensions are all topological dimensions rather than vector-space dimensions, since the sets we're looking at aren't vector subspaces of R^(n^2), but there's nothing abusive about that :-).
It can't be 2n-2 because it's 3 when n=3. I get 2n-3 because the first vector is chosen with n-1 degrees of freedom, then the second with n-2, then subtract one because of the equivalence class of rotations, then add one for choosing how far to rotate.
EDIT: More generally, I think that the dimension of k-dimensional subspaces of an n-dimensional spaces is k(n-k), so where k=2 you get 2n-4, then add one for choosing how far to rotate. I'd feel better if I knew what I meant by "dimension" here though; it's not a vector space.
I recently flipped through the "Cartoon Guide to Physics", expecting an easy-to-understand rehash of ideas I was long familiar with; and that's what I got - right up to the last few pages, where I was presented with a fairly fundamental concept that's been absent from the popular science media I've enjoyed over the years. (Specifically, that the uncertainty principle, when expressed as linking energy and time, explains what electromagnetic fields actually /are/, as the propensity for virtual photons of various strengths to happen.) I find myself happy to try to integrate this new understanding - and at least mildly disturbed that I'd been missing it for so long, and with an increased curiosity about how I might find any other such gaps in my understanding of how the universe works.
So: what's the biggest, or most surprising, or most interesting concept /you/ have learned of, after you'd already gotten a handle on the basics?