A method for estimating the Shannon differential entropy of multidimensional random variables using independent samples is described. The method is based on decomposing the distribution into a product of the marginal distributions and the joint dependency, also known as the copula. The entropy of marginals is estimated using one-dimensional methods. The entropy of the copula, which always has a compact support, is estimated recursively by splitting the data along statistically dependent dimensions. Numerical examples demonstrate that the method is accurate for distributions with compact and non-compact supports, which is imperative when the support is not known or of mixed type (in different dimensions). At high dimensions (larger than 20), our method is not only more accurate, but also significantly more efficient than existing approaches.