Volume form on a manifold is alternating and this lead to impossibility on integration on a non-orientable manifolds (because there is no continuous valume form on them)
But if we consider absolute value of volume form we can construct it continuous easily. why don't we do so?!
We really do so!
Wikipedia:
The absolute value of a "volume form" is a "volume element" which is also known variously as a "twisted volume form" or "pseudo-volume form". It also defines a measure, but exists on any differentiable manifold, orientable or not.