We describe mixed multilinear identities of degree $ 3 $ on right endomorphs of arbitrary algebras over a field $ F $ of characteristic not equal to $ 2 $ .As a consequence, we obtain irreducible bimodules over $ M_{n}(F) $ in the variety defined by the monoassociativity identity and in the variety of $ (1,1) $ -algebras.We construct a broad class of right-symmetric bimodules, including irreducible right-symmetric $ M_{n}(F) $ -bimodules.We introduce a class of $ \omega $ -right-symmetric algebras $ {\mathcal{A}}_{\omega} $ with an $ \omega $ -identity, which generalizes the class of right-symmetric algebras, where $ \omega:{\mathcal{A}}\times{\mathcal{A}}\to F $ is a bilinear skew-symmetric form on $ {\mathcal{A}} $ .We also describe the structure of finite-dimensional algebras $ {\mathcal{A}}_{\omega} $ , in particular, simple algebras of this kind.We prove that the commutator algebra $ {\mathcal{A}}^{(-)} $ of an arbitrary $ \omega $ -right-symmetric algebra $ {\mathcal{A}} $ is an $ \omega $ -Lie algebra and that $ {\mathcal{A}}^{(-)} $ is solvable of degree $ \leq 3 $ in the finite-dimensional case.
