According to Theorem 11.1 in Joseph Lauer's paper, A new length estimate for curve shortening flow and low regularity initial data, the curve shortening flow is well defined and unique for all rectifiable curves in the plane. In particular, triangles do indeed flow to a unique point in their interior.

This construction and related references have also been discussed recently in answer to another question.

@ Willie Wong: Thank you for this reference. Constructing a curve with a prescribed tangential spherical image is the simplest example of convex integration theory which goes back to Whitney. The center of mass trick to ensure that the curve closes up is also mentioned on p. 168 of the book "Partial differential relations" of Gromov.

I just edited my answer above to clarify different interpretations of the term "skew", and corresponding answers in each case.

@Sebastian Goethe: Sorry, I interpreted "skew" as nonparallel. It is not possible for a closed curve in $R^3$ to have nonintersecting tangent lines, but one can construct these in $R^4$. This is described in the paper with Serge Tabachnikov mentioned above.