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 $\delta$-ball 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 $\delta$-balls have the same volume.