dai 带 means to bring sth to some where,a bag,a file~ and dai 戴 means to wear a hat,glasses on your body~ both of them pronounce the same.

No matter what the speaker's intend to say "今天早上我忘记 dai 帽子", it always means " 带 " to the listeners if they don't see the hat with you. Since both 带and 戴 pronounce the same, but in different meanings. If you don't have the hat with you, it is impossible to wear it. Here, your intend is to say that you forget to bring (带) the hat. However, if you have the hat with you then your intend to say is that you forget to wear( 戴) the hat.

If ONLY from your example, I can hardly distinguish the exactly meaning even I am a Chinese....it depends on the context...I think...

For 戴，it should be on your head , around your neck. But for 带，it might be in your bag or your pocket. 我带了帽子．（It might be in my bag OR on my head) 我戴了帽子．（It should be on my head.)

i think that depends the circumstance of the particular case...if you plan to go for a picnic with your family, and you wanna to take a hat to shelter from the sun but you forgot that...then your family members would be curious about that and ask you, did you forget to bring (带）with you? but gernally speaking, we only use 戴 in daily life speaking, like you used to have a hat with you, but today you dont have one... then your collegue would probably ask you , did you forget to wear a hat? 你忘了戴帽子吗？