Chemistry - Are there any other elements that get less dense in their solid state?

Your terminology is a little off: water isn't an element.

There isn't really a simple single explanation of why some solids are less dense than their associated liquids. The general explanation (which doesn't really explain much) is that the solid has a structure and, sometimes, that structure takes us more space than the average structure that occurs in the liquid.

In water, ice has a definite structure where each oxygen sits at the approximate centre of a tetrahedron with two bonds to hydrogen and two hydrogen bonds to the hydrogens of other water molecules. In liquid water the structure is far more fluid and averages out to be slightly more dense that the (more ordered) solid. Exactly why this is true in this case requires a lot more than could fit into an answer here. If water was frozen extremely quickly you might be able to freeze the structure in the liquid; normal cooling allows enough time for the bonding to become more organised.

But the phenomenon isn't unique to water, though the specifics will be very different for other substances. Among elements Gallium, for example, also expands on freezing (don't store it in glass bottles: see this other answer here).

Other examples of elements are germanium and silicon. Acetic acid also forms a less dense solid. In all cases the substances form a relatively ordered crystal structure which just happens to be less dense than the (less ordered) liquid phases.