Is there any other way to test for food allergies besides anelimination diet and skin test?

Yes. Blood testing for food allergy can also be helpful. If a specific food is suspected and confirmed by either skin test or blood test, this may be enough. However, a blinded food challenge is the "gold standard" for food allergy testing.
Those are best. Small amounts of allergic antibody can cause a food allergy and a positive skin test, but may not be enough for a blood test to be positive. The GI tract has its own immune system to suppress most allergic reactions, so even a positive skin test may not mean true allergy. The only way to diagnose food allergy for certain is to remove the food from the diet, and watch for symptom improvement.
Blood testing. Blood tests are not as accurate as skin tests. The best test is an oral challenge but there are risks with that method. Elimination diets are like challenge tests. Elimination diets require eliminating the food(s) in question an then adding the food(s) back in slowly.
Diet alone. Since 95% of food allergies are due to 9 foods - dairy, egg, wheat (oat & barley), soy, peanut, tree nuts, sesame, corn & seafood - simply avoid all of these foods 2 weeks. If you dramatically improve - marginal improvement doesn't count - you can add back 1 food at a time to find the real culprit(s). Among those with food allergy 50% are allergic to 1 food; 99% to no more than 3 foods. Good luck.

Related Questions

Any new way to test for food allergies besides an elimination diet or skin test?

RAST (Immunocap) Rast (aka immunocap) finds the same ige antibodies directed against food allergens as does skin testing. Rast is useful in patients who can't stop taking antihistamines & in patients who have severe eczema, dermatographism, or are very hairy. Igg rast or elisa has not been scientifically shown to be useful although it has its advocates. All tests are surrogates for elimination & oral challenge. Read more...