[Edited for for punctuation, grammar and clarity -- PLC]

Let $M$ be a smooth manifold.  We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric on the ambient space.  

Can all Riemannian metrics on $M$ be constructed in the second way above?