To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which is an $End(E)$-valued 2-form. In each chart $U_\alpha$, $\Omega$ may be described by a $r\times r$ matrix ($r$ rank of $E$) whose entries are 2-forms. The matrices change when the charts changes, but due to the tensoriality and the nature of $\Omega$, some quantities such as the trace or the determinant do not change for overlaping charts. (See, for instance, the first chapter of Lecture Notes on Seiberg-Witten Invariants)

Now to take full advantage of these invariant quantities (= to define Chern classes) one considers powers of the curvature matrix and their traces:

$$\Bigl(\frac{i}{2\pi}\Omega_\alpha\Bigr)^k\qquad\text{tr}\Bigl[\Bigl(\frac{i}{2\pi}\Omega_\alpha\Bigr)^k\Bigr]$$

The procedure is ok, but if one studies it closer, one realizes that we are indeed defining some 'pseudo-wedge' map

$$\Omega^p(End(E))\times\Omega^q(End(E))\longrightarrow\Omega^{p+q}(End(E))$$

by simply taking the product of matrices whose entries are forms. But the question is

**Is there any way to define this pseudo-wedge product intrinsically, that is, by using only the classical wedge product $\Omega^p(M)\times\Omega^q(M)\longrightarrow\Omega^{p+q}(M)$ together with some linear algebra?** Perhaps there is already some book making an explicit definition; in this case it would be most helpful for me to have good references.

Any idea or suggestion is welcome.

EDIT: See the discution below about the definition of Wikipedia and the relationship with the curvature of connections in vector bundles.

Topology, Geometry and Gauge fields: Foundationsin section 5.11 around page 323. $\endgroup$ – Willie Wong Mar 17 '15 at 12:15