# Skew symmetric matrix rows property

If I have any skew symmetric matrix $$\widehat{v}$$ with elements

$$\begin{bmatrix} 0 & -v_{3} & v_{2} \\ v_{3} & 0 & -v_{1} \\ -v_{2} & v_{1} & 0 \end{bmatrix}$$

where $$v\in \mathbb{R} ^{3}$$ then what can we say about the rows of this matrix? Are they linearly independent? The span of these 3 row vectors( Not columns) will be a plane?

Assuming $$v\neq 0$$, $$0$$ as an eigenvalue of the matrix has multiplicity $$1$$ so its eigenspace (which is the same as the nullspace of the matrix) has dimension $$1$$. Therefore, the matrix has (row) rank $$2$$ and the rows will span a plane.