By Girard's Spherical Excess Formula, a spherical triangle on unit sphere with angles $A, B, C$ has area $$ A + B + C - \pi. $$ I would like to know, if there is a generalization for this formula to higher dimensions.

There is a thread on MathOverflow about calculating the volume of elliptical tetrahedron. It seems that for odd dimensions the problem is much complex. But one answer in that thread suggests that it is easier to get one for even dimensions. However I could not find any result through Google.

I really know very little about geometry. Any information regarding this problem would be appreciated.