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Shall I construct a stochastic process $X(t)$ such that $X(s+t)-X(s)\sim U(-t,t)$ ? Or is there already any similar formula?

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Let $U$ be uniform in $[-1,1]$ and let $X_t=Ut$, which is uniform in $[-t,t]$. Then $$X_{t+s}-X_s=U(t+s)-Us=Ut$$ so this works. So it's not so much a stochastic process as just a random variable giving the slope of a line.

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