This question is not elementary given the accepted answer (it is incorrect) and the number of votes it has. You can permit a random slope on extraversion - or any between subject regressor, and there may be information in the data to estimate it.
Douglas Bates has a relevant quote on the reason for such misconceptions in his book on mixed models (http://lme4.r-forge.r-project.org/book/, chapter 2):
The blurring of mixed-effects models with the concept of multiple, hierarchical levels of variation results in an unwarranted emphasis on “levels” when defining a model and leads to considerable confusion. It is perfectly legitimate to define models having random effects associated with non-nested factors. The reasons for the emphasis on defining random effects with respect to nested factors only are that such cases do occur frequently in practice and that some of the computational methods for estimating the parameters in the models can only be easily applied to nested factors.
This is not the case for the methods used in the lme4 package. Indeed there is nothing special done for models with random effects for nested factors. When random effects are associated with multiple factors exactly the same computational methods are used whether the factors form a nested sequence or are partially crossed or are completely crossed.
One of the limitations of the older formulation related to an unwarranted emphasis on levels that became known as truths for mixed effects models is, "you can't have a random slope on a between subjects (level 2) regressor". For example, the software HLM forces you to define the level of variables and does not allow you to do this.
With modern computational approaches, simply think of variables and whether you expect the effect of a variable to vary across the grouping factor, regardless of whether the variable itself varies across the grouping factor. The only reason to worry about levels would be interpretation issues (that social scientists sometimes care about), and predictor specification issues (that they care less about).