I have already proved that there exists a vector isomorphism $f$ between $\mathbb{R}^3$ and the set of $3\times 3$-antisymmetric matrix $\mathfrak{o}(3)$, that satisfies $f(u\times v) =[f(u),f(v)]$ where $\times$ denotes the vector cross product. I wonder if I can deduce something interesting of the Lie algebra $\mathfrak{o}(3)$ from the fact that there is such isomorphism with $\mathbb{R}^3$?
Any idea is welcome.