Here's the definition of $\textit{orientable manifold}$ in Spanier AT
A connected $n$-manifold $X$ is said to be orientable if its homology tangent bundle is orientable [i.e, if there exists an element $U\in H^n(X\times X,X\times X-\delta(X);R)$ such that for all $x\in X, U\mid x\times(X,X-x)$ is a generator of $H^n(x\times(X,X-x);R)$]. (here, $\mid$ stands for the restriction)
But as far as I know, the orientation of manifold is defined by local charts as Wikipedia https://en.wikipedia.org/wiki/Orientability says. Are they equivalent definitions? There's some similar definition on [Homology and the orientability of general manifolds] section in Wikipedia but I'm not sure it's same as Spanier's definition.