Finding a particular form that orients a k-manifold

Question

Suppose one has a $k$-manifold given by $f^{-1}(0)$ for some $C^1$ map $f: U\to \mathbb{R}^{n-k}$ (where $[D f(x)]$ is surjective). How can one construct a form-field $\omega$ that orients the manifold?

In particular I would like to use the fact that $$\Sigma(\vec{v_1},\dots,\vec{v_k}):=\text{sgn}\det[\Delta f_1(x),\dots,\Delta f_{n-k}(x),\vec{v_1},\dots,\vec{v_k}].$$

Using this orientation, how can one construct a form field? In the case of $f(x)=x^2+y^2-1=0$, with $f^{-1}(0)$ the circle, I know one can take $\vec v=[dx,dy]^T$and solve $\det[\Delta f,v]=2xdy+2ydx$, which is a form that orients the circle.

In general, say if one has a 2-manifold, how does this scale?

EDIT: My tentative answer is taking the $v_i$ to be $[dx_1,\dots,dx_n]^t$ and doing the same, but it's not clear to me whether this works in higher dimensions.

EDIT: I'm most interested in the elementary approach, but would be happy to hear how something like the Hodge star operator ties in if there is an explicit relationship.

Answer 1

Do you know about the (Hodge) star operator? This is a great way to get from an ($n-k$)-form to a $k$-form. This will accomplish what you've sketched.

Answer 2

Does this in fact trivially follow from what I already showed? If we take the $v_i$ to all be $[dx_1,\dots,dx_n]^T$, then $\det[Δf_1(x),…,Δf_{n−k}(x),v_1,…,v_k]$ is certainly a form-field.

The only remaining question is whether this always has a consistent sign.