The IKKT matrix model (or the type IIB matrix model) is known as a promising candidate for a nonperturbative formulation of superstring theory in ten dimensions. As a most attractive feature, the model admits the emergence of (3+1)-dimensional space-time associated with the spontaneous breaking of the (9+1)-dimensional Lorentz symmetry. Numerical confirmation of such a phenomenon has been attempted for more than two decades. Recently it has been found that the sign problem, the main obstacle in simulating this model, can be overcome by the complex Langevin method. It has been shown that the Lorenzian version of the model is smoothly connected with the Euclidean version, in which the SO(10) symmetry is found to be spontaneously broken to SO(3). Here we propose to add a Lorentz invariant ``mass'' term to the original model and discuss a scenario that (3+1)-dimensional expanding space-time with Lorentzian signature appears at late times. Some numerical results supporting this scenario are presented.