Does cross product have an identity?

Does cross product have an identity? I.e. Does there exist some $\vec{id}\in \mathbb{R}^3$ such that $$\vec{id} \times \vec{v} = \vec{v}\times \vec{id} = \vec{v} $$ for all $\vec{v}\in \mathbb{R}^3$?

The answer must be no because $\vec{id}\times \vec{v}$ is perpendicular to both $\vec{id}$ and $\vec{v}$ and the only vector that is perpendicular to itself is the $0$ vector. Thus $\vec{id}\times \vec{v}=\vec{v}$ iff $\vec{v}=\vec{0}$ no matter what $\vec{id}$ is, so this cannot be true in general.

Note that for any potential identity vector $\vec u$, we have $$ \vec u \times \vec v + \vec v \times \vec u = \vec 0 $$ for any vector $\vec v$.

Perhaps even easier. Suppose such a vector $\vec{id}$ exists. First note that $\vec{id} = 0$ does not work, so $\vec{id} \ne 0$.

Applying the desired property with $\vec{v} = \vec{id}$ we get $$\vec{id} \times \vec{id} = \vec{id}$$ By antisymmetry, any vector cross itself is 0. So this is a contradiction.