Under what condition is a coordinate chart orientation-preserving

Question

Let $\phi: U \rightarrow O$ be a coordinate chart of an $n$-dimensional differentiable oriented manifold where $U$ is an open subset of the manifold and $O$ is an open subset of $\mathbb{R}^n$. If we see $\phi$ as a diffeomorphism, we can consider whether it is orientation-preserving, i.e. whether the orientations of $U$ and $O$ are consistent with each other. But since $U$ is a subset of a general manifold, how can we know whether its orientation is consistent with that of $O$? For example, if $O$ has the standard orientation, how can we know whether the orientation of $U$ is also the standard one?