All perfect praise be to Allah, The Lord of The Worlds, and may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
A wig is part of a woman`s ornament that she wears for her husband; therefore; it is prohibited to show it to non-Mahrams (Marriageable men) since Allah, The Almighty, Said in this regard: " And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what (must ordinarily) appear thereof; " {An-Nur/31}.
In conclusion, it is prohibited for the Muslim woman to wear a wig instead of the Islamic head cover, and a Muslim should fear Allah in all his/her affairs, and whoever leaves something for Allah`s Sake, he will be rewarded with that which is better. And Allah Knows Best.