A determination of the single W spin density matrix (SDM) elements in the reaction e(+)e(-) -> W+W--> l nu q (q) over bar (l=e/mu) is reported at centre-of-mass energies between 189 and 209GeV. The data sample used corresponds to an integrated luminosity of 520pb(-1) taken by DELPHI between 1998 and 2000. The single W SDM elements, rho(W +/-)(tau tau') (tau,tau'=+/- 1 or 0), are determined as a function of the W- production angle with respect to the e(-) beam direction and are obtained from measurements of the W decay products by the application of suitable projection operators, Lambda(tau tau'), which assume the V-A coupling of the W-boson to fermions. The measured SDM elements are used to obtain the fraction of longitudinally polarised Ws, with the result: sigma(L)/sigma(tot) = 24.9 +/- 4.5 (stat) +/- 2.2 (syst)% at a mean energy of 198 GeV. The SDM elements are also used to determine the triple gauge couplings Delta g(1)(Z), Delta kappa(gamma), lambda(gamma) and g(4)(Z), (kappa) over tilde (Z) and (lambda) over tilde (Z). For the CP-violating couplings the results of single parameter fits are: g(4)(Z) = -0.39(-0.20)(+0.19) (kappa) over tilde (Z) = -0.09(-0.05)(+0.08) (lambda) over tilde (Z) = -0.08 +/- 0.07. The errors are a combination of statistical and systematic errors. All results are consistent with the Standard Model.