The construction via degeneracy loci definitely goes through in the general setting of a complex vector bundles over a smooth compact oriented manifold.

It is often referred to as Gauss-Bonnet type formulae.

You can find a nice account about that in "Principle of Algebraic Geometry" by Griffiths and Harris, Chapter 3, Section 3.

The construction you want via degeneracy loci definitely goes through in the general setting of a complex vector bundles over a smooth compact oriented manifold.

It is often referred to as Gauss-Bonnet type formulae.

You can find a nice account about that in "Principle of Algebraic Geometry" by Griffiths and Harris, Chapter 3, Section 3.