Let $\Omega^2(\mathbb{R}^3)$ represent the collection of differential 2-forms on $\mathbb{R}^3$. For this space we take as an (ordered) basis $\{dx \wedge dy, dx \wedge dz, dy \wedge dz\}$.
First question: Is there a principle, other than convention, that this is the "canonical" or "usual" basis of $\Omega^2(\mathbb{R}^3)$ in terms of the way it is ordered?
Next, let $\{e_1, e_2, e_3 \}$ denote the usual ordered Euclidean basis for $\mathbb{R}^3$. Since the dimension of $\Omega^2(\mathbb{R}^3)$ is $3$, it is isomorphic to $\mathbb{R}^3$ and an isomorphism is determined by assigning $dx$ to $e_1$, $dy$ to $e_2$ and $dz$ to $e_3$ and extending by linearity.
So, here's my primary question: It seems to me that some of these choices are really arbitrary. For instance, we could also construct an isomorphism by assigning $dx$ to $e_2$, $dy$ to $e_1$ and $dz$ to $e_3$ instead. Is there an overriding principle that can be invoked to construct a canonical isomorphism between $\Omega^2(\mathbb{R}^3)$ and $\mathbb{R}^3$? Note also that a similar question applies to an isomorphism between $\Omega^1(\mathbb{R}^3)$ and $\mathbb{R}^3$ since these spaces are also isomorphic.
Update: Based on feedback in the comments, I will try to clarify my question. I am considering differential forms and the alternating spaces in question are over the ring of smooth, real-valued functions. Similarly, by the space $\mathbb{R}^3$ I mean the set of all linear combinations of $fe_1 + ge_2 + he_3$ where $f, g, h$ are smooth real-valued functions. I think though that if we were just considering the algebra of alternating forms, essentially the same questions would apply, yes?