It is known ([1], [3]) that every real Brownian motion $B(t)$, $t\in [0, 1]$, can be represented as $$B(t)=\sum\limits_nZ_n\int_0^tg_n(s)ds$$ where $\{Z_n\}$ is a sequence of i.i.d. symmetric Gaussian random variables, $\{g_n\}$ a CONS in $L^2[0,1]$ and the series is convergent with probability one uniformly over $[0, 1]$. The aim of the present paper is to prove some complete analogues if this fact for Banach space valued Brownian motions.