We use three different cosmological dark matter simulations to study how the orientation of the angular momentum (AM) vector in dark matter haloes evolve with time. We find that haloes in this kind of simulations are constantly affected by a spurious change of mass, which translates into an artificial change in the orientation of the AM. After removing the haloes affected by artificial mass change, we found that the change in the orientation of the AM vector is correlated with time. The change in its angle and direction (i.e. the angle subtended by the AM vector in two consecutive time-steps) that affect the AM vector has a dependence on the change of mass that affects a halo, the time elapsed in which the change of mass occurs and the halo mass. We create a Monte Carlo simulation that reproduces the change of angle and direction of the AM vector. We reproduce the angular separation of the AM vector since a lookback time of 8.5 Gyr to today (alpha) with an accuracy of approximately 0.05 in cos(alpha). We are releasing this Monte Carlo simulation together with this publication. We also create a Monte Carlo simulation that reproduces the change of the AM modulus. We find that haloes in denser environments display the most dramatic evolution in their AM direction, as well as haloes with a lower specific AM modulus. These relations could be used to improve the way we follow the AM vector in low-resolution simulations.