If the null hypothesis is true the p-value is distributed uniformly in [0 1] (at least ignoring discrete data, composite hypothesis and other cases which are no

If the null hypothesis is true the p-value is distributed uniformly in [0 1] (at least ignoring discrete data, composite hypothesis and other cases which are not easy). The distribution will be different when then true hypothesis is not true (but you may also get non-significant results even if the null hypothesis is not true). I’m not sure if that’s what you mean by “the actual distribution will be skewed in cases where it is statistically significant.”