Constructing higher degree non-trivial bounded coho- mology classes is a very challenging task. For surface groups and free groups, bounded cohomology is very rich in degree 2 and 3, and a natural question is whether one can build non-trivial classes in higher degrees by taking the cup product of lower-dimensional classes. For hyperbolic manifolds, there exists a well defined map Ψ• associating to every closed differential form a bounded coho- mology class via integration over straight simplices. Classes in the image of this map are usually called De Rham classes, and, in de- gree 2, they span an infinite-dimensional subspace of the bounded cohomology space of the manifold. We prove that, in suitable degrees, Ψ• is a homomorphism of al- gebras, i.e. it sends the wedge product of closed differential forms to the cup product of the associated bounded cohomology classes. As a corollary, the cup product of two De Rham classes vanishes, provided that its degree exceeds the dimension of the manifold. This result complements several recent vanishing results for the cup product of De Rham classes.