Tangent space of $\mathfrak{ so}(3)$ Lie algebra

Question

Very basic question and the terminology makes it difficult to find a reference. I just know the basics of differential geometry but my question is simple.

Is the tangent space at the point $P\in\mathfrak{so}(3)$ (where $\mathfrak{so}(3)$ is the Lie algebra) equal to the Lie algebra itself?

Thanks!

Answer 1

This follows from the more general fact that the tangent space at a point in a finite-dimensional real vector space $V$ is canonically isomorphic to $V$ itself. The isomorphism associates to $v\in V$ the directional derivative at the point in the direction of $v$.