The covariance matrix tabulates pair-wise interactions between variables (assets) one-at-a-time into a grid, which can quickly become large as the number of assets included in a portfolio, for example, is increased to the hundreds or thousands, contributing to the curse of dimensionality. Elongating the covariance matrix like this also often just merely bumps up the number of corresponding rows and columns that are deemed redundant due to eventually numerous collinearities. If the covariance matrix is fundamental to many multivariate financial models, it is more of a necessity, due to lack of better alternative measures, and far from an ideal.
Like how total correlation does for correlation, is there a measure that represents the entire covariance structure as a scalar or something similarly small? If so, would the pair-wise mentality found in finance prevail regardless because its intuitiveness overrides its issues mentioned above?