A map from a manifold to a projective space correspondent to a subalgebra

Question

Let $M$ be a smooth manifold of dimension $n$ and let $\mathfrak g$ be a finite dimensional subspace of $\Gamma(M,TM)$. Define $V_{\mathfrak g}:= \wedge^n\mathfrak g $. Let $N+1:=\dim V_{\mathfrak g}$ and $<\sigma_0,\dots,\sigma_N>$ be a base of $V_{\mathfrak g}$.

Is the following map well-defined and smooth? $$\phi : M\rightarrow \mathbb P^N(\mathbb R)$$ $$x\mapsto[\sigma_0(x),\dots ,\sigma_N(x)]$$