A woman who does not have a Mahram to travel with does not have to go for Hajj or `Umrah and she is excused if she does not go. It is prohibited for her to travel for Hajj or for any other reason without a Mahram, and she has to be patient until Allah makes it easy for one of her Mahrams to travel with her.

It is permissible for a woman, if she is in Makkah, to go walking from the hotel to the Haram without a Mahram, and to do Tawaf likewise, because it is only necessary to have a Mahram for travelling; inside the city it is not essential to have a Mahram.

It is not permissible for a woman to travel to seek knowledge without a Mahram. She should acquire the knowledge that she needs in the many ways that are available such as listening to tapes, asking scholars over the phone and other means that Allah has made available in these times.

Conclusion: your brother may be your mahram for the purpose of travel if he is trustworthy. We ask Allah, may He be exalted, to guide him. And Allah knows best.