Given 4 points on $S^3$. If we look at the spherical polygon formed on $S^3$, is there a formula for the 3-dimensional Hausdorff measure for it?

E.g.: When I tried to set up a spherical coordinate for the following 4 points, it's not clear to me what is the domain of the spherical angles: $\vec{0}$, $\vec{e_1}$, $\vec{e_2}$, $\vec{e_2}$, $(0,0,1/\sqrt{2},1/\sqrt{2})$.

How about higher dimensions? Thanks!