Mathematically for every pseudovector $x_i$ there is a matrix $F_{ij}=\epsilon_{ijk}x_k$ such that the matrix transforms properly under all orthogonal transformations. Therefore I would expect it would be more natural to write physical quantities such as angular momentum $\textbf{L}$ or magnetic field $\textbf{B}$ in terms of their corresponding matrices.

Is there any physical insight as to why these quantities behave the way they do apart from experimental verification. If it is simply the way they are, is there any insightful interpretation of their corresponding matrices? I seem to be able to get some intuition from looking at the vectors but none at all by analysing the matrices.

Is the way these vectors physically behave related to their pseudoness? For example the rather odd direction of magnetic force.

Mathematically for every 3D pseudovector $x^i$ there is a 2-form $F_{ij}=\epsilon_{ijk}x^k$ such that the 2-form transforms properly under all orthogonal transformations. Therefore I would expect it would be more natural to write physical quantities such as angular momentum $\textbf{L}$ or magnetic field $\textbf{B}$ in terms of their corresponding 2-forms.

Is there any physical insight as to why these quantities behave the way they do apart from experimental verification. If it is simply the way they are, is there any insightful interpretation of their corresponding 2-forms? I seem to be able to get some intuition from looking at the vectors but none at all by analysing the 2-forms.

Is the way these vectors physically behave related to their pseudoness? For example the rather odd direction of magnetic force.