In the field of spatial extremes, stochastic processes with upper semicontinuous (usc) trajectories have been proposed as random shape functions for max-stable models. In the literature dealing with usc processes, max-stability is defined via a sequences of scaling constants, rather than functions, only. It is however not clear whether and how extreme-value theory (EVT) for continuous processes extends to usc processes. In particular, classical multivariate and continuous EVT relies on the probability integral transform and Sklar's theorem. This theorem justifies working with standard marginal distributions, simplifying the task of constructing and characterizing max-stable processes and their domains of attraction. In the present work, we investigate the possibility to follow these steps for usc processes. Unfortunately, the pointwise probability integral transform is not necessarily `permitted': without additional assumptions, the obtained process may not even have usc trajectories. We give sufficient conditions for marginal standardization to be possible, and we state a partial extension of Sklar's theorem for usc processes, with a particular focus on max-stable ones.