PLEASE tell me you're joking...
The EPS shell on the inside of a helmet is specially designed to absorb most of the impact of a fall, allowing as little of the force of the fall as possible to act on your skull. This reduction in the amount of force acting directly on your skull, and in turn, your brain, greatly decreases the rider's risk of sustaining a serious head injury.
A hat, on the other hand, is just cushy and soft.
But to the OP, some people are just cold when they ski, but it is never a good idea to leave your face uncovered on the hill in sunny conditions unless you're using strong sunblock. You can get a really terrible sunburn because the snow is so reflective. Usually, I wear a bandana around my head and one around my face, or I tie a T-shirt around my face (which works really well).
And as for the goggles, goggles won't fall off, they keep wind out of your eyes, and they are far less likely to break; and if they do break, they are far less likely to stab your eyes, as someone above me said.