Context. Planet-forming disks are fundamental objects that are thought to be inherited from large scale rotation through the conservation of angular momentum during the collapse of a prestellar dense core. Aims. We investigate the possibility for a protostellar disk to be formed from a motionless dense core that contains nonaxisymmetric density fluctuations. The rotation is thus generated locally by the asymmetry of the collapse. Methods. We study the evolution of the angular momentum in a nonaxisymmetric collapse of a dense core from an analytical point of view. To test the theory, we performed three-dimensional simulations of a collapsing prestellar dense core using adaptative mesh refinement. We started from a nonaxisymmetrical situation, considering a dense core with random density perturbations that follow a turbulence spectrum. We analyzed the emerging disk by comparing the angular momentum it contains with the one expected from our analytic development. We studied the velocity gradients at different scales in the simulation as is done with observations. Results. We show that the angular momentum in the frame of a stellar object, which is not located at the center of mass of the core, is not conserved due to inertial forces. Our simulations of such nonaxisymmetrical collapse quickly produce accretion disks at the small scales in the core. The analysis of the kinematics at different scales in the simulated core reveals projected velocity gradients of amplitudes similar to the ones observed in protostellar cores and for which directions vary, sometimes even reversing when small and large scales are compared. These complex kinematics patterns appear in recent observations and could be a discriminating feature with models where rotation is inherited from large scales. Our results from simulations without initial rotation are more consistent with these recent observations than when solid-body rotation is initially imprinted. Lastly, we show that the disks that formed in this scenario of nonaxisymmetrical gravitational collapse grow to reach sizes larger than those that are observed, and then fragment. We show that including a magnetic field in these simulations reduces the size of the outcoming disks and it prevents them from fragmenting, as is shown by previous studies. Conclusions. We show that in a nonaxisymmetrical collapse, the formation of a disk can be induced by small perturbations of the initial density field in the core, even in the absence of global large-scale rotation of the core. In this scenario, large disks are generic features that are natural consequences of the hydrodynamical fluid interactions and self-gravity. Since recent observations have shown that most disks are significantly smaller and have a size of a few tens of astronomical units, our study suggests that magnetic braking is the most likely explanation. The kinematics of our model are consistent with typically observed values of velocity gradients and specific angular momentum in protostellar cores. These results open a new avenue in which our understanding of the early phases of disk formation can be explored since they suggest that a fraction of the protostellar disks could be the product of nonaxisymmetrical collapse, rather than directly resulting from the conservation of preexisting large scale angular momentum in rotating cores.