Of course: any classical expression, involving numbers becomes an expression involving operators in quantum mechanics. (The ordering ambiguity requires some more discussion.) So the angular momentum operator in quantum mechanics is the cross product of the position and momentum operators, for instance. Here the ordering ambiguity is easily resolved, since the commutator of position and momentum is proportional to the identity.

As an application you can see the antisymmetric exchange in the interaction between two neighboring magnetic spins, Si and Sj. It can be written as H=Dij . (Si x Sj) . This term is alled Dzyaloshinskii-Moriya interaction where H is the interaction Hamiltonian, Dij is propotional to ri x rj (the position of two ions or spins etc). The spin states in this Hamiltonian are in crossed product form.