Conformal structure on a Riemannian manifold

It is known that on any 2-dimensional Riemannian manifold there exists
a conformal structure induced by the Riemannian metric. This is usually
done by choosing the so-called isothermal coordinates. The question is:

Can any compact Riemann surface be obtained as a 2-dimensional
submanifold in R^3, complex structure being induced by the the
Riemannian metric of R^3?

If not as a surface in R^3 then maybe as a 2-dimensional submanifold of
some R^n?

Ignat Soroko

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