Let's define by $\eta(d,n)$ the number of distinct configurations of $n$ lines in $d$ dimensional space in a way that they all pass through the origin and the angle between any pair of lines is the same as any other pair. Here, distinct means that we shouldn't be able to take one configuration and superpose it on another with a simple rotation.
For example, $\eta(3,4) \geq 2$ because you can take the body diagonals of a cube or those of an Icosahedron and remove two.
Of course, we can always take two lines and place them at any angle to each other, so the configurations must have at least $3$ lines.
Is there a way to calculate this in closed-form or otherwise estimate it for arbitrary $d$ and $n$.
An obvious extension is the series $\gamma(d) = \sum\limits_{n=d+1}^\infty \eta(d,n)$, which defines the total number of non-trivial distinct configurations in $d$ dimensional space.
EDIT: It was pointed out in the comments that $\eta(d,n) = \infty$ for $1<n\leq d$. So the question is interesting when $n>d$.
Related: How to draw six lines in 3-d space such that angle between any pair of them is the same.