See @lee2013smooth proposition 8.11 (exercise)
Proposition
Let $M$ be a smooth $n$-manifold with or without boundary and let $(X_1,\dots,X_k)$ be a linearly independent $k$-tuple of smooth vector fields on an open subset $U$ of $M$, with $1\le k< n$. Then for each $p\in U$ there exist smooth vector fields $X_{k+1},\dots,X_n$ in a neighborhood $V$ of $p$ such that $(X_1,\dots,X_n)$ is a smooth local frame for $M$ on $U \cap V$.
$\blacksquare$
In other words, we can complete to a frame.
I add that if the manifold $M$ is contractible space then the completion can be performed in the whole $M$. Since in this case the tangent bundle is trivial,
$$ g: TM \to M\times \mathbb{R}^n $$we can apply a linear transformation pointwisely
$$ h: M\times \mathbb{R}^n \to M\times \mathbb{R}^n $$such that $h\circ g$ sends $X_i$ to $(0,0,1,0,\ldots,0)$. The desired vector fields are given by the preimage (pre-pushforward, better said) of the rest of the canonical vector fields.
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Author of the notes: Antonio J. Pan-Collantes
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