Skewness decays with time, but the rate of that skewness decay will vary based on the instruments and how they are traded, so a simple estimator such as the square root of time rule is not appropriate.
I typically recommend that to scale VaR or ES it makes more sense to lower your confidence level (raise the alpha parameter) to one that makes sense for your holding period.
So, for example, assume that we are working with daily returns, as in your question. Now assume that I have a one-month holding period, and I want a 'monthly VaR'.
I would argue that a rational confidence level for this is 95%, or 1 in 20, corresponding approximately to the loss that will be exceeded about 1 day a month.
For monthly returns, as in a hedge fund portfolio, a confidence of 92% may be most appropriate, to specify the VaR that will be exceeded, on average, once a year.
I think that this is a much more rational approach than asking 'what loss level will be exceeded once in 10000 years?', as many papers and standards bodies recommend. These numbers aren't very useful, as many other authors have pointed out.
Also, extension to Cornish Fisher Expected Shortfall (also called CVaR or Expected Tail Loss) with the same approach as above, helps scale these numbers in a rational way, asking what the mean loss is when the loss exceeds the VaR.
More information on this is available in our published work including this paper from the Journal of Risk which also covers additive/coherent portfolio decomposition of Cornish Fisher VaR and Expected Shortfall.