Are all metric tensors of 3 dimensional surfaces 2x2?

Question

I was doing some work with the volume surface area ratio when I worked it out, I had assumed the metric tensor to be 3x3. I looked for some examples of surfaces test on trecs.se but I saw that all of the first fundemental forms of the surfaces in the surface index were 2x2. Am I dealing with a small minority where this is true or is there some way to introduce a third parameter keeping the surface the same?