In this paper we consider elliptical random vectors X in R(d), d >= 2 with stochastic representation ARU, where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R(d) and A is an element of R(dxd) is a given matrix. Denote by parallel to.parallel to the Euclidean norm in R(d), and let F be the distribution function of R. The main result of this paper is an asymptotic expansion of the probability P {parallel to X parallel to > u} for F in the Gumbel or the Weibuli max-domain of attraction. In the special case that X is a mean zero Gaussian random vector our result coincides with the one derived in Husler et al. (2002) [1].