We know given a Connection 1-form $\omega$ on a Principle bundle $P(M,G)$ we can define a curvature 2-form $\Omega$ of $\omega$. We also know that given a Connection 1-form $\omega$ we can define a Connection $\lambda$ (a smooth distribution on $P$) .
My question is the following:
Is there any equivalent notion of Curvature 2- form which arise from the notion ( when Connection is given as a smooth distribution on $P$) and independent of the notion(Connection 1-form)? Since we already know that when a Connection is given as a distribution then we can define the notion of Parallel transport and Holonomy groups independent of the notion of Connection 1-form. Can we some way define the notion of Curvature in terms of Parallel transport and Holonomy groups independent of considering the corresponding connection 1-form of the given Connection so that "this" notion of curvature obtained in this way will be equivalent to the our standard notion of Curvature 2-form of a Connection 1 -form?
I think Ambrose-Singer Holonomy Theorem relates partially the Curvature 2-form of a Connection form and the Holonomy of the corresponding Connection. Can it be used in some way to define alternative independent equivalent notion of Curvature of a Connection?