All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
If the dress that you mentioned in the question is peculiar to Christian women or it is one of their slogans, then it is not permissible for a Muslim woman to wear it even if she removes the sign of the cross. That is because wearing it is a kind of imitating them and there are many Prophetic narrations which prohibit Muslims from imitating non-Muslims or wearing their clothes. Imaam Muslim reported that 'Abdullaah Ibn Umar narrated that the Prophet saw him wearing two clothes dyed in safflower and he told him: "These clothes are peculiar to the non-Muslims so do not wear them." The Prophet also said: "Whoever imitates a people, he is one of them." [Abu Daawood] Abu Uthmaan An-Nahdi narrated that when he was in
Therefore, it is prohibited to wear the clothes that are peculiar to non-Muslims. It is not any clothes that the non-Muslims wear that we are prohibited to wear, as you mentioned in your question, but it is only the clothes that are peculiar to them or are one of their slogans that we are prohibited to wear. It is permissible for a Muslim to wear the usual clothes that are not a slogan of non-Muslims.
Allaah Knows best.