We study the Jacobian determinants J = det(∂fi/∂xj) of mappings f: Ω ⊂ ℝn → ℝn in a Sobolev–Orlicz space W1,Φ (Ω,ℝn). Their natural generalizations are the wedge products of differential forms. These products turn out to be in the Hardy–Orlicz spaces ℌp (Ω). Other nonlinear quantities involving the Jacobian, such as J log |J|, are also studied. In general, the Jacobians may change sign and in this sense our results generalize the existing ones concerning positive Jacobians.