We provide a new tableau model from which one can easily deduce the characters of finite-dimensional irreducible polynomial representations of the special orthogonal group $SO_n(\mathbb{C})$. This model originates from the representation theory of the $\imath$quantum group (also known as the quantum symmetric pair coideal subalgebra) of type $\mathrm{A\!I}$, and is equipped with a combinatorial structure, which we call $\mathrm{A\!I}$-crystal structure. This structure enables us to describe combinatorially the tensor product of an $SO_n(\mathbb{C})$-module and a $GL_n(\mathbb{C})$-module, and the branching from $GL_n(\mathbb{C})$ to $SO_n(\mathbb{C})$.