It is well known that the strong error approximation, in the space of continuous paths equipped with the supremum norm, between a diffusion process, with smooth coefficients, and its Euler approximation with step $1/n$ is $O(n^{-1/2})$ and that the weak error estimation between the marginal laws, at the terminal time $T$, is $O(n^{-1})$. In this talk, we study the $p-$Wasserstein distance between the law of the trajectory of a diffusion process, with linear diffusion coefficient, and its Euler scheme. Using the Komlós, Major and Tusnády construction, we show that this Wasserstein distance is of order $\log n /n$.