Efficient way to define many variables?

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Hi, I want to use a computer to look at some examples of a certain determinant function. This function depends on many variables (like up to 50-100 in some examples), and the only way I know how to define variables at the moment is something like this:

a, b, c = var('a, b, c')

So, this limits me to using no more than 26 variables. I would like to have variables x_{i,j}, y_{i,j}, z_{i,j} which are indexed by a pair of positive integers i,j not exceeding a number n (which will depend on the example I'm doing, e.g. let's say n=10).

Is there a good way to define all these variables and have Sage recognize them?

Also, this determinant is a polynomial in these variables, and many terms will cancel. Is there a way to get Sage to simplify the expression and make these cancellations? It seems I would need to tell it that my variables commute with each other, which I don't know how to do.