In this paper, we study the following $n$-dimensional piecewise smooth differential system
$ \begin{eqnarray*} \ \left\{\begin{array}{ll} \dot{x}_1=x_2+\varepsilon g^+_1({\bf x}),\\ \dot{x}_2=-x_1+\varepsilon g^+_2({\bf x}),\\ \dot{x}_3=\varepsilon g^+_3({\bf x}),\\ \cdots\\ \dot{x}_n=\varepsilon g^+_n({\bf x}),\\ \end{array}\right.x_1\geq0,\quad \quad \left\{\begin{array}{ll} \dot{x}_1=x_2+\varepsilon g^-_1({\bf x}),\\ \dot{x}_2=-x_1+\varepsilon g^-_2({\bf x}),\\ \dot{x}_3=\varepsilon g^-_3({\bf x}),\\ \cdots\\ \dot{x}_n=\varepsilon g^-_n({\bf x}),\\ \end{array}\right.x_1<0, \end{eqnarray*}$
where ${\bf x}=(x_1,x_2,\cdots,x_n)^T$, $0<\varepsilon\ll1$, and $g^\pm_i({\bf x})$, $i=1,2,\cdots,n$ are real polynomials of ${\bf x}$ with degree $m$. By using the first order Melnikov vector function, we obtain the upper bound of the number of periodic orbits bifurcating from the unperturbed system.
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