For any natural numbers $k$ and $l<k$, it is defined the action of the algebraic torus $(\mathbb{C}^*)^{k}$ on $\mathbb{C}P^{N-1}$, where $N=\binom{k}{l}$, which is given as the $l$-symmetric power representation of $(\mathbb{C}^*)^{k}$ in $(\mathbb{C}^*)^{N}$ and the standard action of $(\mathbb{C}^*)^{N}$ on $\mathbb{C}P^{N-1}$. It is well known question about description of toric manifolds arising from this action. In this note we solve this problem for $k=2n$ and $l=n$, that is we describe the orbits of this action and their closures, that is the corresponding toric manifolds. In this context we also discuss the $(\mathbb{C}^*)^{2n}$-action on the complex Grassmann manifolds $G_{2n,n}$ by the Plücker embedding $G_{2n,n}\to\mathbb{C}P^{N-1}$, where $N=\binom{2n}{n}$. The explicit expressions of torus orbit closures we obtain can be further used in description of singularities of toric varieties.