Let R be a unital prime ring.Then every surjective mapΦcompletely preserving commutativity fromRonto Rhas the formΦ=LC°π, where C∈Z (R) is an invertible element andπis a ring isomorphism of R.Let R be a unital involutive ring with involution *.Then every surjective mapΦcompletely preserving skew commutativity fromRonto Rhas the formΦ=LC°π, where C∈Z (R) is an invertible symmetric element andπis a *-ring isomorphism of R.If the maps are unital, the primeness assumption on the rings can be deleted, which gives a characterization of ring isomorphisms (*-ring isomorphisms) of general rings (involutive rings) .Applications to some operator algebras are discussed such as C*-algebras, von Neumann algebras, standard operator algebras on Banach spaces, indefinite self-adjoint standard operator algebras on Krein spaces and symmetric standard operator algebras.For the case of standard operator algebras, the surjectivity assumption on the maps can be weakened to require the range contain all rank-1 idempotents.