In 2d the angle between $2$ vectors were simply through the computation of the inner or outer product, and the "maximum" angle between $m$ vectors could be simply extracted by the maximum(mod by $\pi$). (For example, $v_1=(1,0)$, $v_2=(1/2,1/2)$, and $v_3=(1,1)$, the "maximum" angle between those three vector or the solid angle created by those three vector were $\pi/2$.)
The question became more complicated in $3$ d. The solid angle between $3$ vectors were given by some non trivial formula (How to find out the solid angle subtended by a tetrahedron at its vertex? ), and the "maximum" solid angle of $m$ vectors(mod by half of the solid angle of a ball) does not seem to be able to be simply computed by a scalar $\max$.
How to compute solid angle of $m$ vectors in $n$ dimensional space?
This is not as simple as it may seem and there are no easy generalizations to N dimensions. This predominantly comes from the fact that the solid angle is effectively a surface integral over the N-1 sphere with a boundary that is only piecewise continuous. Here is a paper I found that gives a non-closed form formula found on page 4. https://link.springer.com/content/pdf/10.1007/s00454-006-1253-4.pdf