In this paper we introduce the notion of the osculating space of a minimal surface as a vector bundle formed from the span of the z-derivatives of a conformal parametrization of the surface. We show that a minimal surface being J-holomorphic is equivalent to the dot product being fully degenerate on the osculating space. We then show that any minimal surface lying fully in R^n with osculating space of dimension half the ambient space must either be J-holomorphic or has a complex structure that is not compatible with the Euclidean metric. Considering the dot product as a bilinear form on the osculating space, we prove some strong restrictions on the dimensionality of the null space of this bilinear form. We show that the osculating space behaves nicely with respect to decomposable minimal surfaces and that we can always decompose a minimal surface into a J-holomorphic minimal surface and a minimal surface for which the dot product is non-degenerate on the osculating space. Finally, we prove that complete finite total curvature stable surfaces of genus one in R^5 are holomorphic under the condition that a certain cover of the surface is stable and the normal bundle is not topologically trivial.