In complexity theory, the size of a problem is the number of bits to represent the input, so if the input integer is i, then the size is n = log_2(i) so the algorithm is actually exponential in the number of bits it takes to represent the input.
But in practice, you're not. Case in point: matrix multiplication is commonly quoted as O(n^3) (naive), when in fact, the amount of data used is O(n^2), and therefore should be quoted as O(size^2) (quadratic) with respect to data-size. (EDIT: was bad at math for a second).
But no, people mean "n-by-n matrix" has "O(n^3) naive" implementation. In practice, the "n" is just whatever is most convenient for a problem.
In many cases, n is proportional to the input size. But in other cases (such as the famous matrix multiplication examples), n is the sqrt(input-size).
It's actually O(size^1.5), not O(size^2). Well, assuming you're talking about θ(n) when you mean O(n) as most people do.
[If you're not familiar with the difference, Θ(f(n)) means a function that grows as fast as f(n), up to a constant, whereas O(f(n)) is a function that grows no faster. So n^2 is O(n^3) but not Θ(n^3).]