Quite high. When you compress air it heats up and that energy is lost.

When you expand it it gets very cold and you lose pressure (you can wait a bit for the air to warm it up to regain some of what was lost).

Could one store the excess heat from the compression and use this to heat the de-compressed air later?

It's not easy to store heat. If it was possible (practical) you could skip the compressed air and just store the heat directly and use it for energy.

The whole reason for the complicated schemes is because it's hard to store heat.