Let $X_1,\dots,X_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X_1(p),\dots,X_n(p))$ is a basis for all $p \in \mathbb R^n$.

Question: Is it possible to choose a cube $C$ around the origin of $\mathbb R^n$ such that there is for every $p \in C$ a piecewise smooth curve $\alpha \subset C$ which connects $p$ with $0$ where the smooth parts of the curve are given by the flows of the vector fields $\pm X_1,\dots,\pm X_n$? (With other words: is it possible to travel from $0$ to $p$ following only the integral curves of the given vector fields in a bounded domain?)

For $n=2$ this is pretty clear; w.l.o.g. $X_1=\partial/\partial x_1$ and the image of $x_1=0$ under the flow $X_2$ fills all of $\mathbb R^2$, since the flow lines of $X_2$ intersect $x_1=0$ transversally. Now it is easy to find a $C$ and $\alpha$ for a $p \in C$. But I can not generalize this for $n$ arbitrary.

Let $X_1,\dots,X_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X_1(p),\dots,X_n(p))$ is a basis for all $p \in \mathbb R^n$.

Question: Is it possible to choose a cube $C$ around the origin of $\mathbb R^n$ such that there is for every $p \in C$ a piecewise smooth curve $\alpha \subset C$ which connects $p$ with $0$ where the smooth parts of the curve are given by the flows of the vector fields $\pm X_1,\dots,\pm X_n$? (With other words: is it possible to travel from $0$ to $p$ following only the integral curves of the given vector fields in a bounded domain?)

For $n=2$ this is pretty clear; w.l.o.g. $X_1=\partial/\partial x_1$ and the image of $x_1=0$ under the flow $X_2$ fills all of $\mathbb R^2$, since the flow lines of $X_2$ intersect $x_1=0$ transversally. Now it is easy to find a $C$ and $\alpha$ for a $p \in C$. But I can not generalize this for $n$ arbitrary.

Edit: Instead of demanding $C$ to be a cube, one could also ask if there is a open neighbourhood of the origin with the desired properties.