In some physics applications, I am aware that the first Chern number and the second can be interpreted as the degree of some maps.
For example, the first Chern number appears in topological insulators, e.g. in the Haldane models, where it is the degree of the map from a torus $T^2$ (the first Brillouin zone) to a 2-sphere $S^2$.
A similar interpretation is also found for the 2nd Chern number. In the theory of instantons, i.e. Yang-Mills theory in $R^4$, there is $$ w(g)=\frac{1}{32\pi^2}\int_{S^3}\textrm{Tr}\left[{g^{-1}dg\wedge g^{-1}dg\wedge g^{-1}dg}\right]$$
where $g:S^3\rightarrow SU(2)$ is a map (a gauge transformation), and $w$ is the winding of this map. $g^{-1}dg$ is Maurier-Cartan form. In physics this is called the Pontryagin index, winding number, Instanton action, Wess-Zumino-Witten (WZW) action, etc. This term is also part of the non-Abelian Chern-Simons action.
This term is equal to $$ \frac{1}{8\pi^2}\int_{S^4}\textrm{Tr}\left[F\wedge F\right] $$ where $F = dA+A\wedge A$ is the curvature, $A$ is the gauge potential (local connection form). This is the second Chern number.
Question:
I am not sure if the interpretation of Chern numbers as the degree of maps is true for all Chern numbers in general. In other words, are the above just coincidences? Is there a universal interpretation such that the $n$th Chern number can be interpreted as the degree of some maps from some $M$ to some $N$, the structure of which depend on $n$? Is such interpretation important (in the sense that it is a common perspective in learning Chern classes)?