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This paper develops a normative approach to the measurement of ex-ante inequality of opportunity in a multidimensional setting - that is, when the individual outcome is represented by a multidimensional variable. We characterize three classes of social welfare functions, all endorsing ex-ante compensation but each reflecting a specific reward principle: (1) utilitarian, (2) agnostic, and (3) averse. The first class is implemented via generalized Lorenz dominance applied to each attribute separately. The agnostic and inequality-averse classes are implemented by a welfarist Lorenz ordering, namely of type-aggregate utilities. In the case of the inequality-averse class, utility functions are submodular, hence capturing the dependence between attributes. We also develop normative inequality indices for the classes of welfare functions and study their properties. Finally, we propose an empirical application of the methods developed in the paper. By using the National Longitudinal Study of Adolescent to Adult Health, we evaluate inequality of opportunity in the USA for the case of three dimensions of individual outcomes: education, health, and income.