There has been [work][1] on [Gram matrices][2] that appears relevant, see e.g. Theorems 14-5 on p. 24-5.

Also of peripheral interest: there is a [hyperbolic generalization][3] of the [Dehn invariant][4]. But as far as I can tell this sort of thing can't really be a generalization of any 2D construction.


  [1]: http://homepages.warwick.ac.uk/~masgar/Seminar/WGTS/Abrosimov.pdf
  [2]: https://en.wikipedia.org/wiki/Gramian_matrix
  [3]: https://lamington.wordpress.com/2009/07/28/brianchon-gram-sommerville-and-ideal-hyperbolic-dehn-invariants/
  [4]: https://en.wikipedia.org/wiki/Hilbert%27s_third_problem