Yes, it is men who are technically designated in the Bible to be the leaders and “rulers”. But that sort of leadership and rulership doesn't mean that women should serve men. To the contrary:
Matthew 20:25-28 “The rulers of the Gentiles lord it over them, and their superiors exercise authority over them. It shall not be this way among you. Instead, whoever wants to become great among you must be your servant, and whoever wants to be first among you must be your slave— just as the Son of Man did not come to be served, but to serve, and to give His life as a ransom for many.”
Eph 5:25,28-29 “Husbands, love your wives, just as Christ loved the church and gave himself up for her…28 So husbands ought also to love their own wives as their own bodies. He who loves his own wife loves himself; 29 for no one ever hated his own flesh, but nourishes and cherishes it, just as Christ also does the church.”
Men are supposed to love, nourish, and cherish their wives like Christ loved, nourished, and cherished the church. That’s a lot of responsibility. Because how did Christ exhibit His love to the Church? Christ expressed His love by becoming a servant to them (see Matt 20:28) and sacrificing everything:
2 Corinthians 8:9 "For you know the grace of our Lord Jesus Christ, that though He was rich, yet for your sakes He became poor, so that you through His poverty might become rich."
Even his own life. (See, again, Eph 5:25)
The Bible also states that men should submit to their wives just like women should submit to their husbands. (Eph 5:21) So, yeah. Biblically speaking, men are supposed to serve their wives.