Given an $n$-tuple $(x_1,\dots,x_n)$ of unit vectors in $\mathbb R^n$, the probability in question is 
$$p_n:=2P(X_1>0,X_j<0\ \forall j=2,\dots,n),$$
where $X_j:=x_j\cdot G$ for $j=1,\dots,n$ and $G$ is a standard Gaussian random vector in $\mathbb R^n$, so that the covariance matrix of $(X_1,\dots,X_n)$ is the Gram matrix of $(x_1,\dots,x_n)$. 

Schläfli (1858) obtained a differential recurrence relation for probabilities of the form $p_n/2$; see e.g. [Plackett][1] and references there. Using this recurrence for $n\le4$, $p_n$ can be expressed as an integral of elementary functions. 
As pointed out by [Ruben. p. 213][2],
"For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer be expressed in terms of elementary functions". 


  [1]: https://www.jstor.org/stable/2332716?seq=1
  [2]: https://www.jstor.org/stable/2333017?seq=1