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Here is a definition that works for all dimensions, including zero.
Let $V$ be a real vector space equipped with a Riemannian metric.

An orientation on $V$ is an isometric isomorphism between $\mathbb R$ and the top exterior power of $V$.

If you want to get rid of the Riemannian metric, you can also say that an orientation is a ray inside the top exterior power of $V$.

Here is a similarly well-behaved definition of spin structure. It also works all the way down to dimension zero. As above, let $V$ be a real vector space equipped with a Riemannian metric.

A spin structure on $V$ is a Morita equivalence between Cliff($\mathbb R^n$) and Cliff($V$). In other words, it is a Cliff($\mathbb R^n$)-Cliff($V$)-bimodule that has the property that it induces an equivalence between the category of Cliff($\mathbb R^n$)-modules and that of Cliff($V$)-modules

To see the analogy with the above definition of orientation, note that $\Lambda^{top}\mathbb R^n=\mathbb R$. And so requiring an isomorphism between $\mathbb R=\Lambda^{top}\mathbb R^n$ and $\Lambda^{top}V$ is formally similar to requiring a Morita equivalence between Cliff($\mathbb R^n$) and Cliff($V$). You just replace the functor $\Lambda^{top}$ by the functor Cliff, and replace the 1-category of vector spaces with the 2-category of algerbas and bimodules.

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Here is a definition that works for all dimensions, including zero.
Let $V$ be a real vector space equipped with a Riemannian metric.

An orientation on $V$ is an isometric isomorphism between $\mathbb R$ and the top exterior power of $V$.

If you want to get rid of the Riemannian metric, you can also say that an orientation is a ray inside the top exterior power of $V$.