You need to be careful beucase the wedge product can be used in on any forms in the exterior algebra but the cross product only works in three dimientions.

The more general idea is

The Hodge dual is a map from

Loosely speaking If we are given an r form w we are looking for a 3-r form v

such that ()

Where J is the Jacobian matrix (the volume element of the curvilinear coordinate system.)

In rectangular coordinates |J|=1

So in your first case you have

2nd

Notice the the order is wrong we need to move dx to the front and since the wedge product is antisymmetric we get a factor of (-1) each time we move through a 1-form this gives

So the Hodge dual is

and for the last one I assume you have a typo and that you mean

So the Hodge dual is