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Let R′=GR(ps,psml) and R=GR(ps,psm) be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis for R′ over R, where R′ is considered as a free module over R. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis for R′ over R and the set of all invertible, circulant, and orthogonal matrices over R.