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 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, "For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer be expressed in terms of elementary functions".

Note that the random vector $G/\|G\|$ is uniformly distributed on the unit sphere in $\mathbb R^n$, where $G$ is a standard Gaussian random vector in $\mathbb R^n$.

So, given an $n$-tuple $(x_1,\dots,x_n)$ of unit vectors in $\mathbb R^n$, the probability that the random hyperplane separates $x_1$ from $x_2,\dots,x_n$ is $$p_n:=2P(X_1>0,X_2<0,\dots,X_n<0),$$ where $X_j:=x_j\cdot G$ for $j=1,\dots,n$, 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 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, "For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer be expressed in terms of elementary functions".