An analytical expression for the von Neumann entropy of the Laughlin wave function is obtained for any possible bipartition between the particles described by this wave function, for a filling fraction . Also, for a filling fraction , where is an odd integer, an upper bound on this entropy is exhibited. These results yield a bound on the smallest possible size of the matrices for an exact representation of the Laughlin ansatz in terms of a matrix-product state. An analytical matrix-product state representation of this state is proposed in terms of representations of the Clifford algebra. For , this representation is shown to be asymptotically optimal in the limit of a large number of particles.