Hardy Weinberg principle
You almost said it all! Here are two ways to think of this problem

Finding the missing probability

Let's denote the fraction yz genotype in the population with $f(yz)$. The sum of of fraction of all genotypes must be equal to 1. In equation it means

$$f(aa) + f(aA) + f(AA) = 1$$

Here, of course I don't make a difference between $f(aA)$ and $f(Aa)$ (whether A is inherited from the mother or from the father). Knowing as you said

$$f(aa) = p^2$$

and 

$$f(AA) = q^2$$

and feeding this value back to the first equation result into 

$$p^2 + f(aA) + q^2 = 1$$

Solving for $f(aA)$ yields to

$$f(aA) = 2pq$$

Directly calculate the probability

Another to reach this result is to ask the following questions


  What is the probability that the maternal allele is A?


The answer is $q$


  What is the probability that the paternal allele is a?


The answer is $p$


  What is the probability that the maternal allele is a?


The answer is $p$


  What is the probability that the paternal allele is A?


The answer is $p$

Therefore, $$f(aA) = pq + pq = 2pq$$