Partial derivatives on a manifold at point $P$ can be thought of as equivalence classes of curves which pass through $P$ with the same velocity. Likewise, first-order differential forms ($dx$, $dy$, you know), which are dual to partial derivatives, can be thought of as equivalence classes of scalar fields whose gradients at $P$ are the same. This provides an extremely natural way to construct the inner product between a partial derivative and a differential form: It is simply the derivative with respect to the parameter of the composition of the parametric curve with the scalar field, $\frac{d}{dt} f(s(t))$, evaluated at $P$.
That's nice, and it helps ground the topic, but are there any similar representations that may be constructed for higher-order differential forms, like $dx \wedge dy$?