We know that the Vandermonde determinant of order $n$ is the determinant defined as follows:

$$\begin{vmatrix} 1&x_1&x_1^2&\dots&x_1^{n-1}\\ 1&x_2&x_2^2&\dots&x_2^{n-1}\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 1&x_n&x_n^2&\dots&x_n^{n-1}\\ \end{vmatrix}=\prod\limits_{1\leq i<j\leq n}(x_j-x_i).$$

For any or some special nonempty subset $S\subseteq \{(i,j)\mid1\leq i<j\leq n\}$，does there exist some matrix $A_S$ which is similar to the Vandermonde matrix (for example, every element of $A_S$ is something just like ${x_i}^j$) such that $$\begin{vmatrix} A_S \end{vmatrix}=\prod\limits_{(i,j)\in S}(x_j-x_i)$$ $$\text{or}$$$$\begin{vmatrix} A_S \end{vmatrix}\neq 0 \quad\text{if and only if}\prod\limits_{(i,j)\in S}(x_j-x_i)\neq 0\quad?$$