We present a generalized notion of extreme multivariate dependence between two random vectors which relies on the extremality of the cross-covariance matrix between these two vectors. Using a partial ordering on the cross-covariance matrices, we also generalize the notion of positive upper dependence. We then quantify the strength of dependence between two given multivariate series using an entropic distance to extremally dependent distributions. We apply this method to build indices of exposure to a financial environment, and to do stress-tests on the correlation between two sets of financial variables.