Let $K$ be a set of all Killing vector fields on $\mathbf R^n$ (with the Euclidean metric $\bar g$) which vanish at the origin.
(A vector field $V$ on a Riemannian manifold $(M, g)$ is said to be a Killing vector field if the flow of $V$ acts by isometries of $M$. This is equivalent to saying that $\mathcal L_Vg=0$).
If $V\in K$, then by using $\mathcal L_V\bar g=0$, we get that the matrix $[\partial V^i/\partial x^j]$ is anti-symmetric, where $V^i$ are the components of $V$ in the standard coordinates.
Define a map $T:K\to \mathfrak o(n)$ as
$$T(V)= \left[\frac{\partial V^i}{\partial x^j}(0)\right]$$ where $\mathfrak o(n)$ is the Lie algebra of $O(n)$, which is "same" as the space of $n\times n$ real anti-symmetric matrices.
Problem. To show that $T$ is injective.
I am quite lost here.