Generalization of the Heckscher-Ohlin Model
The HO model has been generalised. Vanek does a good job of it. 

Instead of only two countries, there is an index of countries.


There are many industries.
Identical technology
Identical, homothetic tastes.


The HOV theorem states that if a country is abundant in a factor,
its factor content of trade in that factor should be positive, and
negative otherwise.

Empirically, this model is not that successful. Here is a good paper discussing the applications and results.

See below for that extension:
Vanek, Jaroslav, The Factor Proportions Theory: The N-Factor Case,” Kyklos, October 1968, 21, 749-755.