In this talk we will investigate how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases as an extension of the known method of matrix representation by ONBs. We will give basic definitions of the functions connecting infinite matrices defining bounded operators on l2 and operators on H. We will show some structural results and give some examples. Furthermore in the case of Riesz bases we prove that those functions are isomorphisms. We are going to apply this idea to the connection of Hilbert-Schmidt operators and Frobenius matrices. Finally we will use this concept to show that every bounded operator is a generalized frame multiplier.