The geometry of the Lie algebra of the orthogonal group $O(\mathbb R^4)$

In the $6$-dimensional space $\Lambda_2(\mathbb R^4)$ of bivectors a Lie product is introduced analogous to the standard vector product in $\mathbb R^2$. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the group of orthogonal transformations $O(\mathbb R^4)$. This isomorphism of Lie algebras is a canonical isometry of the space of antisymmetric operators in $\mathbb R^4$ onto $\Lambda_2(\mathbb R^4)$.