Air can most definitely be felt! Haven't you ever been sat in a draught or been blown about in a wind?
The sensation of 'touch' (which is what you're referring to a 'feeling') relies upon the skin sensing an abnormal pressure. In still air, everything is 'normal' and hence nothing is 'felt'. It's only when the air moves that the pressure on the skin is increased and it's felt.
You can only see (directly) something which reflects light into the eye. (You can also perceive the presence of something, even though you're not actually 'seeing' it, simply because it blocks light from entering the eye). Since air allows light to pass through it (i.e. air neither reflects nor blocks light) you truly can't see it. However the presence of air can be detected (which, if you like, can be taken as 'seen') when it refracts light. The sky only appears blue because of the refraction of light in the air so, in a way, you're 'seeing' air.
Chris