Computing the exponential of large-scale skew-Hermitian matrices or parts thereof is frequently required in applications. In this work, we consider the task of extracting finite diagonal blocks from a doubly-infinite skew-Hermitian matrix. These matrices usually have unbounded entries which impede the application of many classical techniques from approximation theory. We analyze the decay property of matrix exponentials for several classes of banded skew-Hermitian matrices. Then finite section methods based on the decay property are presented. We use several examples to demonstrate the effectiveness of these methods.