How do I denote arbitrary element of these vectors?

You can technically do whatever you want as long as it’s comprehensible (actually, it doesn’t even need to be comprehensible, but if it isn’t, nobody will care—just look at some of Galois’s manuscripts!) but what you might find useful is $v_{i,j}$ which may denote the $i$-th component of the $j$-th vector, or vice-versa.

When I have to manipulate indices of many vectors, I like calling them $\{v^{(1)},...,v^{(m)}\}$ and labeling elements of those vectors $v^{(i)}_j$ because I prefer to keep vector indices downstairs when I can. You can also drop the parantheses. Whatever makes most sense to you is good, as long as other people can understand it as well.