The C*-algebra U_{nc}(n) is the universal C*-algebra generated by n^2 generators u_{ij} that make up a unitary matrix. We show that Kirchberg's formulation of Connes' embedding problem has a positive answer if and only if U_{nc}(2) \otimes_{\min} U_{nc}(2)=U_{nc}(2) \otimes_{\max} U_{nc}(2). Our results follow from properties of the finite-dimensional operator system V_n spanned by 1 and the generators of U_{nc}(n). We show that V_n is an operator system quotient of M_{2n} and has the OSLLP. We obtain necessary and sufficient conditions on V_n for there to be a positive answer to Kirchberg's problem. Finally, in analogy with recent results of Ozawa, we show that a form of Tsirelson's problem related to V_n is equivalent to Connes' embedding problem.