Whoops! The second link was meant to go to Oscar Cunningham's blog post: oscarcunningham.com/494/…

It gives you all of $\mathbb{R}_{\geq 0}$. Rationals correspond to sequences that are eventually zero.

Have you looked at Noam Elkies's Mordell-Weil lattices? They're defined for power-of-two dimensions and they're record-breakingly dense in the range you describe: citeseerx.ist.psu.edu/viewdoc/… There's a sequel of this paper which looks specifically at the dimension-128 lattice. I don't know whether anyone has tried to solve the closest vector problem in these lattices, though.

(In the generalisation where $k$ can be any positive integer, not just $k = \binom{n}{2}$)

@MichaelHardy In terms of choosing $c_1, \dots, c_{\binom{n}{2}}$ to give the good $\varepsilon$-approximability, that's effectively a separate question unrelated to $SO(n)$. For two values, $1, \phi$ is known to be optimal, but I'm unsure how this generalises to higher dimensions.

@MichaelHardy $\binom{n}{2} = \frac{1}{2}n(n-1)$ is already the number of elements strictly above the (all-zero) diagonal.

No, $C$ is a 12-dimensional orthoplex (24 vertices) inscribed in a 12-dimensional hypercube (4096 vertices). The codimension-1 configuration $C'$ is an 11-dimensional simplex (12 vertices) inscribed in an 11-dimensional hypercube (2048 vertices).