Show that the groups $\text{GL}(N,R), \text{GL}(N.C), \text{SL}(N,R), \text{SL}(N,C), O(N), SO(N), U(N),SU(N)$ have dimension $N^2, 2N^2, N^2-1, 2(N^2-1), \frac{1}{2}N(N-1), \frac{1}{2}N(N-1), N^2$ and $N^2-1$ respectively.
Attempt:
Elements in $\text{GL}(N,R)$ are just $N \times N$ matrices with real components so such a matrix has $N \cdot N = N^2$ free parameters. For $\text{GL}(N,C)$ each matrix element is of form $x+iy$ so constitutes two free parameters per element and thus $2N^2$ overall. Restricting to $\text{det}=1$ means we can choose $N^2-1$ elements free and the last one is fixed so that the determinant condition is fulfilled. This gives the answers to the first four. Is this ok?
Now a member of $O(N)$ can be written as $\exp(iA)$ where $A$ must be an antisymmetric matrix to guarantee the orthogonality condition is satisified by this representation of an element of $O(N)$. An antisymmetric matrix has $N(N-1)/2$ independent components. Is this an ok argument? Why is it that $SO(N)$ has the same dimension? Intuitively should it not change in an analogous way that the det condition had to be fulfilled for the special linear groups?
Many thanks!