The volume in the orthogonal group is measured by the Haar measure, which is the up to scaling unique measure that is invariant under the group operation. I consider the usual metric that is induced by the spectral norm M = max Mx where x ranges over all vectors of length 1 and the vector norm is the Euclidean one. A \deltaball is the set of all orthogonal matrices that have distance less or equal \delta to a fixed matrix M. Because of the invariance of the Haar measure, for a fixed \delta, all \deltaballs have the same volume.

The volume of the deltaball of the special orthogonal group can be computed exactly by applying the Weyl integration formula: (Without loss of generality, we assume that the deltaball is around the unit group element). a. One notices (Again due to the invariance under the Haar measure) that the characteristic function of the delta ball is a class function. Thus upon the application of the Weyl integration formula we are left only with the radial part on the eigenvalues which is a $\lfloor N/2\rfloor$dimensional integral for $\mathrm{SO}(N)$. Here, the radial integral is described explicitely. b. The eigenvalues of an orthogonal matrix of dimension $N=2m+1$ are $1$ and $m$ pairs $\exp(i \phi_ m)$ and $\exp(i \phi_ m)$, $0\leq\phi_ 1 \leq\ldots\leq\phi_m \leq\pi$. In the case of even dimensions, the unit eigenvalue is absent. c. The deltaball condition on the eigenvalues becomes: $$ \exp(i\phi_k)1\leq\delta , $$ which implies: $$\phi_k\leq 2 \arcsin\sqrt{\delta/2}.$$ d. Applying the Weyl integration formula, we obtain for the odd case $\mathrm{SO}(2m+1)$: $$ \mathrm{Vol}(\delta\mathrm{ball}) = \frac{2^{m^2}}{\pi^m m!} \int_{\phi_1\leq\ldots\leq\phi_m \leq 2 \arcsin\sqrt{\delta/2}} \prod_{1\leq j < k \leq m} (\cos\phi_k\cos\phi_j)^2 \prod_l \sin^2(\phi_l) d\phi_1 \cdots d\phi_k. $$ e. For the even dimensional case, the only changes are $2^{m^2}$ is replaced $2^{(m1)^2}$ and the sine terms are absent. 

