We study the elephant random walk in arbitrary dimension $d\geq 1$. Our main focus is the limiting random variable appearing in the superdiffusive regime. Building on a link between the elephant random walk and P\'olya-type urn models, we prove a fixed-point equation (or system in dimension two and larger) for the limiting variable. Based on this, we deduce several properties of the limit distribution, such as the existence of a density with support on $\mathbb R^d$for$d\in\{1,2,3\}$, and we bring evidence for a similar result for$d\geq 4$. We also investigate the moment-generating function of the limit and give, in dimension $1$, a non-linear recurrence relation for the moments.