Detect which antigen binds to IgE
From what I understand most allergies are detected by a scratch test. from the link below:

"A skin prick test, also called a puncture or scratch test, checks for immediate allergic reactions to as many as 40 different substances at once. ... After cleaning the test site with alcohol, the nurse draws small marks on your skin and applies a drop of allergen extract next to each mark. He or she then uses a lancet to prick the extracts into the skin's surface. A new lancet is used for each allergen. ...About 15 minutes after the skin pricks, the nurse observes your skin for signs of allergic reactions. If you are allergic to one of the substances tested, you'll develop a raised, red, itchy bump (wheal) that may look like a mosquito bite"

More information on scratch tests can be found here:

http://www.mayoclinic.org/tests-procedures/allergy-tests/basics/what-you-can-expect/prc-20014505

After a quick google search,I found a paper from 2014 [1] that seems to speak to the essence of what you are asking, link below.

http://www.tandfonline.com/doi/abs/10.1586/14737159.4.4.539 [1]

In the paper above they propose using a protein microarray technology to find allergies. 

I am not sure if there is any way to extract the igE-antigen complex from the patient. But I think it could probably be done with the microarray technology. Although it would be cool to see a picture of the complex. 
I hope I have helped answer your question.

[1] Harwanegg, Christian, and Reinhard Hiller. "Expert Review of Molecular Diagnostics." Taylor and Francis Online. Expert Review of Molecular Diagnostics, 09 Jan. 2014.