  In this paper, we study the Donsker scaling limit of integer-valued random
walks perturbed on a finite subset of $\mathbb Z$ called a membrane. Under very
mild assumptions about the law of the random walk's increments inside and
outside of the membrane we show weak convergence of the scaled processes to a
skew Brownian motion and provide the explicit formula for its permeability
parameter in terms of stationary distributions of certain embedded Markov
chains. The proof is based on a representation of the original random walk as a
multidimensional coordinate process and its convergence to a Walsh Brownian
motion.
