How to Represent Set S to Distinguish a Single n from All Natural Numbers?

[SprucerMoose]

G'day all,

If i let S={[a_ij]_nxn, where a_ij = a_ji, n\inZ⁺}

How do I distinguish between a single natural number n or all natural numbers?

I am trying to show that for each n, this set makes a subspace, but am not sure how to represent this set without the ambiguity, as this is not a subspace if the set contains more than a single n.

Thanks

 

[Office_Shredder]

Define the sets
S_n = \{ \left[ a_{ij} \right]_{n\times n}\ |\ a_{ij}=a_{ji} \}
for each n a positive integer. Then S_n is a vector space for each choice of n.

 

[SprucerMoose]

Thank you very much