Gumbel here, clayton there

Copulae can at the same time be both very useful and somewhat impractical.

Useful, because they allow for a more precise modelling of co-variation between variables, empowering our models with the non-symmetry, non-normality found ever so often in the real world.

Impractical, becuse multivariate co-variation can be very hard to model, even worse to parameterize. A good description of A ∩ B can be done, but when you model A ∩ B ∩ C ∩ D ∩ E ..., the fitting is so complex that the previously so well described relation between A ∩ B is no longer precise.

Solution
The solution is closer than it might seem. Keep modelling the pairs, and model the mulitvariate whole as a cascade of two-by-two pairs. A refreshing paper by Aas, Czado, Frigessi, and Bakken was publicized in Insurance: Mathematics and Economics, 2007: Pair-copula constructions of multiple dependence.