Is multivariable maximum a linear combination of absolute values?

Question

There is a well known equality states that for all $a,b\in\mathbb R$, we have $$\max(a,b)=\frac{a+b}{2}+\left|\frac{a-b}{2}\right|. $$ Suppose $n\ge 3$ is an integer, can we find real numbers $a_{ij}\ (0\le i\le m,1\le j\le n),b_i\in\{-1,1\}\ (1\le i\le m)$ such that for all real numbers $x_1,\cdots,x_n$, the following equality always holds? $$\max(x_1,\cdots,x_n)=\sum_{j=1}^na_{0j}x_j+\sum_{i=1}^m b_i\left|\sum_{j=1}^na_{ij}x_j\right|. $$