In Frankel's book he writes that in $R^{3}$ with cartesian coordinates, you can always associate to a vector $\vec{v}$ a 1-form $v^{1}dx^{1}+v^{2}dx^{2}+v^{3}dx^{3}$ and a two form $v^{1}dx^{2}\wedge dx^{3}+v^{2}dx^{3}\wedge dx^{1} +v^{3}dx^{1} \wedge dx^{2}$, now in general this is not possible and you have to convert the components of a vector with a metric to get covariant components, for example like this $v_{i}=g_{ij}v^{j}$
Then he asks what is in general the associated 2-form for $\vec{v}$ ? He then proofs that to a vector $\vec{v}$ one does associate a pseudo-2-form $v^{2}:= \iota_{\vec{v}}vol^{3}$ Later when he discusses the cross product he writes that one would like to say that $v^{1} \wedge \omega^{1}$ is the 2-form associated to the vector $\vec{v} \times \vec{w}$, but we only have a pseudo-2-form associated to a vector thus the pseudovector $\vec{v} \times \vec{w}$ is associated to the 2-form $v^{1} \wedge \omega^{1}$ (which is just a flip flop of words I think).
Now is it true that if we have other coordinates than cartesian, can one only associate pseudo-forms to a vector? Because in the text he calls the forms in cartesian coordiantes just forms, but in general he says pseudo-forms.
Since $v^{2}:= \iota_{\vec{v}}vol^{3}$ the association explicitly depends on a choice of the volume form. Frankel probably doesn't add "pseudo" to forms in $\mathbb{R}^3$ because he thinks of it as having a fixed coordinate system, which determines the standard volume form. Riemannian metric plus orientation will also determine a volume form, as will a symplectic form. But if you allow transformations that do not respect the volume form you only get pseudo.