Yes, the Bible teaches we are to submit to each other. Submitting one to another in the fear of God.* This specific verse is used over and over again to invalidate the wives submitting to husband verses. They say we are all called to submit to each other.
The Bible always explains itself. Right after this verse it teaches what submitting to each other looks like.
Wives, submit yourselves unto your own husbands, as unto the Lord. For the husband is the head of the wife, even as Christ is the head of the church: and he is the savior of the body. Therefore as the church is subject unto Christ, so let the wives be to their own husbands in every thing.*
This is the submission that wives are called upon to do. Our husbands are our heads and we are to submit to them in every thing. They are the leaders of the home and we are the followers.
Husbands, love your wives, even as Christ also loved the church, and gave himself for it:...so ought men to love their wives as their own bodies. He that loves his wife loves himself. For no man ever yet hated his own flesh; but nourisheth and cherisheth it...*
This is the submission that husbands are called upon to do. They are called to love, nourish, and cherish their wives. So you see that submission has a different definition for the husband and the wife. Both roles are equally important. When they are obeyed, there is harmony and beauty in a marriage.
However, just because one of the spouses isn't obeying God in their submissive role doesn't give you the right to disobey your role. If you don't believe your husband is loving, nourishing, and cherishing you, it doesn't give you the right to disobey or disrespect him as your head. You stand responsible to God for your actions and your actions alone.
Remember, God gives this wonderful promise of winning your husband without a word as you live out your biblical submission in front of him. It takes a strong woman to be submissive. To not argue, fight back, and try to get your own way but instead to give up control and lovingly serve and please your husband through God's power working mightily within you takes a strong woman.
Don't believe the lies that society tells you that submission is weakness. Submission is strength and it works. I have witnessed women winning their husbands every time they become submissive. Believe the Truth. Believe God and play out your submissive role in marriage and it just may draw your husband into his role of being submissive to you... Loving, nourishing, and cherishing you.
*All the Bible verses are from Ephesians 5:21-29