The simplest compact Lie groups have familiar volumes. What generalizations are there? For a volume we need a measure on the space. We want the measure to be invariant under the group. For the measure of an infinitesimal volume element near the group identity we take the measure of the corresponding element in the tangent space there. The tangent space can be taken as an n-dimensional Euclidian space with its familiar measure.
The Cartesian product of two measure spaces has a natural measure. So for a fibre bundle whose base and fiber spaces are measure spaces. As the volume of the product space is the product of the volumes of the factor spaces, so is the volume of a volume of fiber bundle space the product of the volumes of the base and fiber spaces.
The special orthogonal Lie group, SOn(R) is a fibre bundle whose base space is S(n−1) and whose fibre is SOn−1(R). Ergo the volume of SOn−1(R) is ∏nγn for r ranging from 1 to n and where γn is the volume of the unit n ball.
γn = πn/2/((n/2)!)
(−½)! = √π;
x! = x((x−1)!)