This paper proposes empirically tractable multidimensional matching models, focusing on worker-job matching. We generalize the parametric model proposed by Lindenlaub (2017), which relies on the assumption of joint normality of observed characteristics of workers and jobs. In our paper, we allow unrestricted distributions of characteristics and show identification of the production technology, and equilibrium wage and matching functions using tools from optimal transport theory. Given identification, we propose efficient, consistent, asymptotically normal sieve estimators. We revisit Lindenlaub's empirical application and show that, between 1990 and 2010, the U.S. economy experienced much larger technological progress favoring cognitive abilities than the original findings suggest. Furthermore, our flexible model specifications provide a significantly better fit for patterns in the evolution of wage inequality.