"Treshold Garch" or T-Garch models are designed to capture this asymmetry. See this exposition by U. Chicago's Ruey Tsay who has a terrific text on time-series models in "Analysis of Financial Time Series".
You can use the structure of the T-Garch models to simulate data with this property.
There is a package called fGarch that creates APARCH models. A T-GARCH model is a special case of an APARCH model where delta = 1. See Ruey's Lecture 5 and associated R-code for using the fGarch library.
Also, there is some outstanding theoretical research by Capital Fund Management using the statistical physics approach on how to explain the negative-skewness and other features. I include a link and abstract to their research below:
More stylized facts of financial markets: leverage effect and
We discuss two more universal features of stock markets: the so-called
leverage effect (a negative correlation between past returns and
future volatility), and the increased downside correlations. For
individual stocks, the leverage correlation can be rationalized in
terms of a new `retarded' model which interpolates between a purely
additive and a purely multiplicative stochastic process. For stock
indices a specific market panic phenomenon seems to be necessary to
account for the observed amplitude of the effect. As for the increase
of correlations in highly volatile periods, we investigate how much of
this effect can be explained within a simple non-Gaussian one-factor
description with time independent correlations. In particular, this
one-factor model can explain the level and asymmetry of empirical
exceedance correlations, which reflects the fat-tailed and negatively
skewed distribution of market returns.