Which basis 2-form elements represent positive traversals in Minkowski 4-space?

I’m certain some of this relies on arbitrary choice, for even in Euclidean 3-space, there is no a priori preferred choice of left versus right hand coordinates. In fact according to Einstein:

There are thus two kinds of Cartesian systems which are designated as
"right-handed" and "left-handed" systems. The difference between these is
familiar to every physicist and engineer. It is interesting to note that these
two kinds of systems cannot be defined geometrically, but only the contrast
between them. https://www.gutenberg.org/files/36276/36276-pdf.pdf page 10, footnote.

One interpretation I have seen for a 2-form basis element $dx^{alpha}wedge dx^{beta}$ is as a traversal around a differential two-surface, first along $dx^{alpha}$ then along $dx^{beta}$ and continuing to complete the circuit. In Euclidean 3-space represented using right-handed rectangular Cartesian coordinates the positive traversals are $dxwedge dy,$ $dywedge dz$ and $dzwedge dx.$ Since the wedge and cross products are synonymous in this context we can express the oriented surface elements as $dxwedge dy=dz,$ etc. Thus $dywedge dx=-dz$ would represent a negative traversal.

In Minkowski 4-space we have the same parity, $dx^{alpha}wedge dx^{beta}=-dx^{beta}wedge dx^{alpha}$. But now we encounter a number of complicating factors. First of all, a 2-surface traversal determines a 2-space orthogonal complement which must be spanned by two basis 1-forms, or by a basis 2-from which is linearly independent of the first basis 2-form.

There is also a question of how the Minkowski metric ${-,+,+,+}$ (per MTW) influences the algebraic sign of a traversal. I will assume that a positive traversal associates with a positively oriented complementary basis two-form.

An example of the use of a two-form basis is in the expression given in Box 4.1 A.2 of MTW for the Faraday two-form:

$$mathbf{F}=-E_{x}dtwedge dx-E_{y}dtwedge dy-E_{z}dtwedge dz$$

$$+B_{x}dywedge dz+B_{y}dzwedge dx+B_{z}dxwedge dy.$$

Is it possible to consistently identify which basis 2-forms represent positive traversals? Is there an established convention? If so what is it?

I will leave the question of Hodge duality of basis 2-forms for the future, unless it is essential to addressing the current question.

As an example in which oriented traversals arise see Box 15.1 of MTW. I may be able to extract a satisfactory answer by carefully re-reading the first 400 pages. But I don’t recall MTW giving a simple listing of the positively oriented two-surfaces of spacetime. The discussion of Hodge duals makes it more difficult to sort this out.