Gauss in his book "Disquisitiones arithmeticae" considered only forms $ax^2+bxy+cy^2$ where $b$ is even, apparently because he had some notion of integral matrix in his mind even though he did not state it explicitly. This restriction has an advantage to present a form in Gauss's form with an integral matrix.

elements of $Sym^2\mathbb{Z}^2$ can be viewed naturally as forms $ax^2+bxy+cy^2$ where $a,b,c\in\mathbb{Z}$. Can we find such a representation like above for Gauss's form i.e some sort of symmetric power of a free module?

This question might be vague, but any interpretation might be useful for my purpose.