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Let $V\subset \mathbb{R}^n$ be a subspace of the standard euclidean space. Is there a constructive method to find a basis of $V$?

Obviously, if we can realize $V$ as the image or kernel of an explicit linear transformation, then there are methods to do this, but the only way I know to find such a linear transformation requires you already have a basis (or at least a spanning set of vectors).

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    what do you know about $V$?? – 2017-02-20
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    In other words, how is $V$ given? If not equations, then perhaps generators? – 2017-02-20
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    I'm asking for an abstract $V$ (which probably means the answer is no). My motivation is that I'm teaching the service linear algebra course and was wondering if I could avoid a non-constructive proof of the existence of a basis. – 2017-02-20

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If $V$ is subset of standard Euclidian space than $V$ can have dimensions 0,1,2 or 3. If $V$ is an euclidean plane then it has dim $2$ and the base is any of two vectors that are nonparallel and lay on that plane. If $V$ is a line, then you can take any vector that is parallel with that line for a base. If $V$ is a point then dim is $0$. In any other cases you have provide additional information about $V$, for example linear operator that describes it and from there you can get vector base.