Variance and skewness are the 2nd and 3rd statistical moments of a random variable's distribution. Unlike the variance-covariance matrix, which is shaped $p\times p$, the skewness-coskewness matrix is shaped $p\times p^2$. Does this make it a rank 3 tensor?
For bivariate data, $p=2$, the co-skewness matrix (flattened tensor?) looks like $$S = \begin{pmatrix} s_{1,1,1} & s_{1,1,2} & s_{2,1,1} & s_{2,1,2} \\ s_{1,2,1} & s_{1,2,2} & s_{2,2,1} & s_{2,2,2} \\ \end{pmatrix} $$ Which of the elements are skewness elements, rather than co-skewness elements?
And how can this matrix be displayed as a tensor?